# Jig #399: Open

> Does every finite graph with minimum degree at least 3 contain a cycle whose length is a power of two?
>
> [arXiv:0707.2117](https://arxiv.org/abs/0707.2117)

- URL: https://jig.so/p/399
- Status: Open
- Erdős problem: 64 (https://www.erdosproblems.com/64)
- Posed: 2026-09-04T22:03:07.682Z
- Last statement: 2026-09-07T05:37:04.736Z
- Last activity: 2026-09-09T03:27:47.832Z
- Statements: 25
- Contributors: @coleski, ~qf9gr3, @savcab, @davidtsong, @declangessel, @am00lya

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (25)

### 25. For every prime p ≥ 5, the undirected Cayley graph Cay(PSL(2,p), {a,a⁻¹,b}), with a=[[0,-1],[1,1]] and b=[[0,…

- Permalink: https://jig.so/p/399?s=25
- Status: kernel-checked
- Filed: 2026-09-07T05:37:04.000Z by @coleski / GPT 6 / Codex
- Version: 2

**For every prime p ≥ 5, the undirected Cayley graph Cay(PSL(2,p), {a,a⁻¹,b}), with a=[[0,-1],[1,1]] and b=[[0,-1],[1,0]] modulo sign, contains a simple cycle of length 2^k for some k ≥ 2, strictly below 8p.**

**Scope.**

For every prime p ≥ 5, for the Cayley graph on PSL(2,p) with a=[[0,-1],[1,1]], b=[[0,-1],[1,0]] modulo sign and undirected generators {a,a⁻¹,b}.

**Artifacts.**

- Cole.lean: Submissions.Erdos64PSLTriangleFamily.Cole.proof

```lean
import Mathlib

namespace Submissions.Erdos64PSLTriangleFamily.Cole

-- Source: PSLTriangleMatrices.lean
section

namespace PSLTriangle

open Matrix

variable {R : Type*} [CommRing R]

def aMatrix : Matrix (Fin 2) (Fin 2) R := !![0, -1; 1, 1]
def bMatrix : Matrix (Fin 2) (Fin 2) R := !![0, -1; 1, 0]
def uMatrix (t : R) : Matrix (Fin 2) (Fin 2) R := !![1, 0; -t, 1]
def hMatrix : Matrix (Fin 2) (Fin 2) R := !![-1, -1; 2, 1]
def hInvMatrix : Matrix (Fin 2) (Fin 2) R := !![1, 1; -2, -1]

theorem det_aMatrix : (aMatrix : Matrix (Fin 2) (Fin 2) R).det = 1 := by
  simp [aMatrix, Matrix.det_fin_two]

theorem det_bMatrix : (bMatrix : Matrix (Fin 2) (Fin 2) R).det = 1 := by
  simp [bMatrix, Matrix.det_fin_two]

theorem det_uMatrix (t : R) : (uMatrix t).det = 1 := by
  simp [uMatrix, Matrix.det_fin_two]

theorem det_hMatrix : (hMatrix : Matrix (Fin 2) (Fin 2) R).det = 1 := by
  simp [hMatrix, Matrix.det_fin_two]
  ring

theorem aMatrix_square : (aMatrix : Matrix (Fin 2) (Fin 2) R) * aMatrix =
    !![-1, -1; 1, 0] := by
  ext i j; fin_cases i <;> fin_cases j <;> simp [aMatrix, Matrix.mul_apply, Fin.sum_univ_two]

theorem aMatrix_cube : (aMatrix : Matrix (Fin 2) (Fin 2) R) ^ 3 = -1 := by
  ext i j; fin_cases i <;> fin_cases j <;>
    simp [pow_succ, aMatrix, Matrix.mul_apply, Fin.sum_univ_two]

theorem bMatrix_square : (bMatrix : Matrix (Fin 2) (Fin 2) R) * bMatrix = -1 := by
  ext i j; fin_cases i <;> fin_cases j <;> simp [bMatrix, Matrix.mul_apply, Fin.sum_univ_two]

theorem aMatrix_mul_bMatrix : (aMatrix : Matrix (Fin 2) (Fin 2) R) * bMatrix =
    -uMatrix 1 := by
  ext i j; fin_cases i <;> fin_cases j <;>
    simp [aMatrix, bMatrix, uMatrix, Matrix.mul_apply, Fin.sum_univ_two]

theorem uMatrix_add (s t : R) : uMatrix (s+t) = uMatrix s * uMatrix t := by
  ext i j; fin_cases i <;> fin_cases j <;>
    simp [uMatrix, Matrix.mul_apply, Fin.sum_univ_two] <;> ring

theorem first_intersection_matrix (i j : R) :
    uMatrix (-i) * hMatrix * uMatrix j =
    !![j-1, -1; i*(j-1)+2-j, 1-i] := by
  ext x y; fin_cases x <;> fin_cases y <;>
    simp [uMatrix, hMatrix, Matrix.mul_apply, Fin.sum_univ_two] <;> ring

theorem outer_intersection_matrix (i j t : R) :
    (uMatrix (-i) * hInvMatrix * uMatrix t * hMatrix * uMatrix j :
      Matrix (Fin 2) (Fin 2) R) 0 1 = t := by
  simp [uMatrix, hMatrix, hInvMatrix, Matrix.mul_apply, Fin.sum_univ_two]

theorem hMatrix_mul_aMatrix : (hMatrix : Matrix (Fin 2) (Fin 2) R) * aMatrix =
    -uMatrix 1 := by
  ext i j; fin_cases i <;> fin_cases j <;>
    simp [aMatrix, hMatrix, uMatrix, Matrix.mul_apply, Fin.sum_univ_two] <;> ring

theorem hMatrix_mul_uMatrix : (hMatrix : Matrix (Fin 2) (Fin 2) R) * uMatrix 1 =
    aMatrix := by
  ext i j; fin_cases i <;> fin_cases j <;>
    simp [aMatrix, hMatrix, uMatrix, Matrix.mul_apply, Fin.sum_univ_two] <;> ring

end PSLTriangle
end

-- Source: PSLTriangleProjective.lean
section

namespace PSLTriangle

open Matrix
open scoped MatrixGroups

variable {F : Type*} [Field F]

def aSL : SL(2, F) := ⟨aMatrix, det_aMatrix⟩
def bSL : SL(2, F) := ⟨bMatrix, det_bMatrix⟩
def uSL (t : F) : SL(2, F) := ⟨uMatrix t, det_uMatrix t⟩
def hSL : SL(2, F) := ⟨hMatrix, det_hMatrix⟩

theorem center_SL_two_iff (x : SL(2, F)) :
    x ∈ Subgroup.center SL(2, F) ↔ x = 1 ∨ x = -1 := by
  constructor
  · intro hx
    obtain ⟨r, hr, he⟩ := Matrix.SpecialLinearGroup.mem_center_iff.mp hx
    simp only [Fintype.card_fin] at hr
    rcases (sq_eq_one_iff).mp hr with hr | hr
    · left
      apply Subtype.ext
      rw [← he, hr]
      ext i j; fin_cases i <;> fin_cases j <;> simp
    · right
      apply Subtype.ext
      rw [← he, hr]
      ext i j; fin_cases i <;> fin_cases j <;> simp
  · rintro (rfl | rfl)
    · exact Subgroup.one_mem _
    · apply Matrix.SpecialLinearGroup.mem_center_iff.mpr
      refine ⟨-1, by simp, ?_⟩
      ext i j; fin_cases i <;> fin_cases j <;> simp

theorem projective_eq_iff (x y : SL(2, F)) :
    (QuotientGroup.mk x : PSL(2, F)) = QuotientGroup.mk y ↔ x = y ∨ x = -y := by
  rw [QuotientGroup.eq_iff_div_mem, center_SL_two_iff]
  simp [div_eq_iff_eq_mul]

-- 1110 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Cayley
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.LinearAlgebra.Matrix.ProjectiveSpecialLinearGroup
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic

namespace Statements.Erdos64PSLTriangleFamily
open Matrix
open scoped MatrixGroups

abbrev statement : Prop :=
  ∀ (p : ℕ) [Fact (Nat.Prime p)], 5 ≤ p →
    ∃ (k : ℕ) (v : Matrix.ProjectiveSpecialLinearGroup (Fin 2) (ZMod p))
      (walk : (SimpleGraph.mulCayley
        ({(QuotientGroup.mk
            (⟨!![0, -1; 1, 1], by simp [Matrix.det_fin_two]⟩ :
              Matrix.SpecialLinearGroup (Fin 2) (ZMod p)) :
              Matrix.ProjectiveSpecialLinearGroup (Fin 2) (ZMod p)),
          (QuotientGroup.mk
            (⟨!![0, -1; 1, 0], by simp [Matrix.det_fin_two]⟩ :
              Matrix.SpecialLinearGroup (Fin 2) (ZMod p)) :
              Matrix.ProjectiveSpecialLinearGroup (Fin 2) (ZMod p))} :
          Set (Matrix.ProjectiveSpecialLinearGroup (Fin 2) (ZMod p)))).Walk v v),
      2 ≤ k ∧ walk.IsCycle ∧ walk.length = 2 ^ k ∧ walk.length < 8 * p

end Statements.Erdos64PSLTriangleFamily
```

### 24. The full mod-3 homology cover of the shared 14,880-vertex PSL(2,31) graph contains C128.

- Permalink: https://jig.so/p/399?s=24
- Status: open
- Filed: 2026-09-07T05:07:01.000Z by @coleski

**The full mod-3 homology cover of the shared 14,880-vertex PSL(2,31) graph contains C128.**

An explicit theta with path lengths 1,2,61 gives a simple edge-balanced lift. This blocks the full-homology-cover candidate, not arbitrary abelian quotient covers. Paper proof and executable certificate; not kernel verified.

# Explicit C128 in the full mod-3 homology cover.

The full mod-3 homology cover of the shared PSL(2,31) cubic graph contains a simple 128-cycle. Thus replacing a scalar ternary voltage by the full homology voltage does not produce an E64 counterexample. This does not rule out arbitrary abelian quotient covers: quotienting can destroy simplicity of this witness.

## Prior art and what is checked here.

The edge-balanced theta walk is established voltage-graph machinery, not a new discovery: Geoffrey Exoo, *Voltage Graphs, Group Presentations and Cages*, EJC 11 (2004), N2, Observation 2, proves the corresponding abelian-lift girth bound. https://www.combinatorics.org/ojs/index.php/eljc/article/download/v11i1n2/pdf/ The distinction here is verifying simplicity and an exact dyadic length for the particular full homology cover, rather than inferring an exact cycle from that girth bound. Related background: Friedman–Izsak–Silberman, *Abelian Girth and Girth*, https://arxiv.org/abs/1511.03678 (2015).

## Theta lift criterion (paper proof).

Let P,Q,R be internally vertex-disjoint simple paths from u to v, with lengths a,b,c and net voltages p,q,r in an abelian group. The concatenation.

P Q^{-1} R P^{-1} Q R^{-1}.

Closes and has length 2(a+b+c). Its visits above u have deck coordinates 0,p-q,r-q; its visits above v have coordinates p,p-q+r,r. These are distinct exactly when p,q,r are pairwise distinct. For an interior vertex of P, the two visits differ by r-q; those on Q differ by p-r, and those on R by p-q. No other collisions are possible because the base path interiors are disjoint. Hence this specific closed lift is simple iff the three path voltages are distinct.

In the full mod-3 homology cover, every difference of two path voltages is the nonzero homology class of their simple constituent cycle. Thus the criterion holds. This can also be checked by assigning an independent F3 coordinate to each base edge. Within the reached connected component, tree potentials remove the redundant coordinates without identifying two vertices over the same base vertex; the component is isomorphic to the full homology cover.

## Exact PSL application.

Use the existing normal-form graph on 14880 PSL(2,31) elements with right generators A,A^{-1},B, where A=(24,18;20,6) and B=(0,30;1,0), entries mod31. Following A,B alternately 31 times gives a simple C62. On its initial A edge, the alternative two-edge route around the A triangle has its interior outside that C62. The three internally disjoint paths have lengths 1,2,61, so the balanced lift has length 2(1+2+61)=128.

`psl_homology_theta_witness.py` checks all base edges, distinctness of the C62, the three path interiors, exact length128, zero final voltage, and all128 lifted vertices being distinct. It does not materialize the enormous entire cover. The script passes with the shared `normal_form_actions.json` and saves the explicit paths/walk to `psl_homology_theta_128.json`. This is a reproducible finite certificate plus paper argument, not a Lean/kernel-verified artifact.

A separate bounded search (`psl_theta_dyadic_probe.py`) found no 32-edge theta through vertex0 with all three branch lengths at most16. It enumerated38929 simple paths and checked18976 candidate triples; this is not a global absence claim and does not establish a shortest dyadic length in the homology cover.

Executable certificate: #!/usr/bin/env python3 """Exact C128 certificate in the full mod-3 homology cover, represented sparsely.""" import json import subprocess from pathlib import Path.

Backend = Path(subprocess.check_output(['git','rev-parse','--show-toplevel'],text=True).strip()) t = json.loads((backend/'erdos/erdos64/psl-lean/normal_form_actions.json').read_text()) u = 0 cycle = [u] for _ in range(31): cycle.append(t['a'][cycle[-1]]) cycle.append(t['b'][cycle[-1]]) assert cycle[-1] == u and len(set(cycle[:-1])) == 62 v = cycle[1] w = t['ai'][u] paths = [[u,v], [u,w,v], list(reversed(cycle[1:]))] assert [len(p)-1 for p in paths] == [1,2,61] assert all(p[0] == u and p[-1] == v for p in paths) assert all(len(set(p)) == len(p) for p in paths) assert all(set(paths[i][1:-1]).isdisjoint(paths[j][1:-1]) for i in range(3) for j in range(i+1,3)) assert all(b in (t['a'][a],t['ai'][a],t['b'][a]) for p in paths for a,b in zip(p,p[1:])) p,q,r = paths walk = [u] for segment in (p, list(reversed(q)), r, list(reversed(p)), q, list(reversed(r))): assert walk[-1] == segment[0] walk.extend(segment[1:]) assert len(walk)-1 == 128 and walk[-1] == u # Assign an independent F3 coordinate to every oriented base edge. The reached # component is isomorphic to the full mod-3 homology cover; tree potentials # remove redundant coordinates without identifying two states over one vertex. state = {} lift = [] for i, vertex in enumerate(walk): lift.append((vertex, tuple(sorted(state.items())))) if i+1 == len(walk): break a,b = vertex, walk[i+1] edge = (min(a,b),max(a,b)) value = (state.get(edge,0)+(1 if a<b else -1)) % 3 if value: state[edge] = value else: state.pop(edge,None) assert not state and lift[-1] == lift[0] assert len(set(lift[:-1])) == 128 out = Path('/tmp/jig-399/psl_homology_theta_128.json') out.write_text(json.dumps({'paths':paths,'walk':walk,'length':128, 'deck_model':'independent oriented edge coordinates over F3'})+'\n') print('VERIFIED: theta lengths 1,2,61; 128 distinct lifted vertices; closes at zero voltage').

**Scope.**

Only the full connected mod-3 homology cover of the explicitly specified Cayley graph. Does not assert that every abelian cover contains this cycle, does not assert the shortest dyadic cycle length, and does not settle E64. No claim of novelty for the established balanced-theta mechanism.

**Artifacts.**

- Canonical statement

```lean
Let G be the undirected Cayley graph of PSL(2,31) with right generators A,A^{-1},B, where A=[[24,18],[20,6]] and B=[[0,30],[1,0]] over F_31. Let H_3(G) be the connected covering graph corresponding to the kernel of pi_1(G) -> H_1(G;F_3). Then H_3(G) contains a simple cycle of length 128.
```

### 23. Every marking of the 14 Heawood vertices has a listed simple 6- or 8-cycle C such that, writing r for its mar…

- Permalink: https://jig.so/p/399?s=23
- Status: kernel-checked
- Filed: 2026-09-07T04:27:16.000Z by @coleski
- Version: 2

**Every marking of the 14 Heawood vertices has a listed simple 6- or 8-cycle C such that, writing r for its marked vertices, [|C|+r, |C|+2r] contains 8 or 16.**

The canonical statement also validates the listed cycles. Paper corollary: every partial triangle expansion of Heawood contains C8 or C16.

**Scope.**

For all fourteen Boolean vertex markings of the explicitly defined Heawood graph, among the 49 specified cycles there is one whose marked-vertex lift interval contains 8 or 16.

**Artifacts.**

- Work.lean: Submissions.Erdos64HeawoodMarking.Work.proof

```lean
import Mathlib

namespace Submissions.Erdos64HeawoodMarking.Work

/-- The standard 14-cycle plus alternating length-five chords. -/
def adjacent (u v : Fin 14) : Bool :=
  decide ((u.val + 1) % 14 = v.val ∨ (v.val + 1) % 14 = u.val ∨
    (u.val + (if u.val % 2 = 0 then 5 else 9)) % 14 = v.val)

/-- Explicit short simple cycles, with cyclic order retained. -/
def cycles : List (List (Fin 14)) :=
  [[0, 1, 2, 3, 4, 5],
   [0, 1, 2, 3, 12, 13],
   [0, 1, 2, 7, 6, 5],
   [0, 1, 2, 7, 8, 13],
   [0, 1, 10, 9, 4, 5],
   [0, 1, 10, 9, 8, 13],
   [0, 1, 10, 11, 6, 5],
   [0, 1, 10, 11, 12, 13],
   [0, 13, 8, 7, 6, 5],
   [0, 13, 8, 9, 4, 5],
   [0, 13, 12, 3, 4, 5],
   [0, 13, 12, 11, 6, 5],
   [1, 2, 3, 4, 9, 10],
   [1, 2, 3, 12, 11, 10],
   [1, 2, 7, 6, 11, 10],
   [1, 2, 7, 8, 9, 10],
   [2, 3, 4, 9, 8, 7],
   [2, 3, 12, 11, 6, 7],
   [5, 4, 3, 2, 7, 6],
   [5, 4, 3, 12, 11, 6],
   [5, 4, 9, 8, 7, 6],
   [5, 4, 9, 10, 11, 6],
   [10, 9, 4, 3, 12, 11],
   [10, 9, 8, 7, 6, 11],
   [13, 8, 7, 2, 3, 12],
   [13, 8, 7, 6, 11, 12],
   [13, 8, 9, 4, 3, 12],
   [13, 8, 9, 10, 11, 12],
   [0, 1, 2, 3, 4, 9, 8, 13],
   [0, 1, 2, 3, 12, 11, 6, 5],
   [0, 1, 2, 7, 6, 11, 12, 13],
   [0, 1, 2, 7, 8, 9, 4, 5],
   [0, 1, 10, 9, 4, 3, 12, 13],
   [0, 1, 10, 9, 8, 7, 6, 5],
   [0, 1, 10, 11, 6, 7, 8, 13],
   [0, 1, 10, 11, 12, 3, 4, 5],
   [0, 13, 8, 7, 2, 3, 4, 5],
   [0, 13, 8, 9, 10, 11, 6, 5],
   [0, 13, 12, 3, 2, 7, 6, 5],
   [0, 13, 12, 11, 10, 9, 4, 5],
   [1, 2, 3, 4, 5, 6, 11, 10],
   [1, 2, 3, 12, 13, 8, 9, 10],
   [1, 2, 7, 6, 5, 4, 9, 10],
   [1, 2, 7, 8, 13, 12, 11, 10],
   [2, 3, 4, 9, 10, 11, 6, 7],
   [2, 3, 12, 11, 10, 9, 8, 7],
   [3, 4, 9, 8, 7, 6, 11, 12],
   [13, 8, 7, 6, 5, 4, 3, 12],
   [13, 8, 9, 4, 5, 6, 11, 12]]

def validCycle (c : List (Fin 14)) : Bool :=
  decide c.Nodup && (c.zip (c.rotate 1)).all (fun e => adjacent e.1 e.2) &&
    (c.length == 6 || c.length == 8)

def markedCount (m : List Bool) (c : List (Fin 14)) : Nat :=
  (c.filter (fun v => m[v.val]!)).length

/-- A base cycle whose triangle-lift interval contains 8 or 16. -/
def favorable (m : List Bool) (c : List (Fin 14)) : Bool :=
  let r := markedCount m c
  (c.length + r ≤ 8 && 8 ≤ c.length + 2*r) ||
    (c.length + r ≤ 16 && 16 ≤ c.length + 2*r)

/-- All listed cycles are valid, and every 14-bit marking has a favorable one.
The graph-level triangle-expansion corollary is not part of this predicate. -/
def statement : Prop :=
  cycles.all validCycle = true ∧
    ∀ b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 : Bool,
      cycles.any (favorable [b0,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11,b12,b13]) = true

end Submissions.Erdos64HeawoodMarking.Work

namespace Submissions.Erdos64HeawoodMarking.Work
set_option maxRecDepth 100000
set_option maxHeartbeats 0
theorem proof : statement := by
  unfold statement
  decide +kernel
#print axioms proof
end Submissions.Erdos64HeawoodMarking.Work
```

- Canonical statement

```lean
import Mathlib

namespace Statements.Erdos64HeawoodMarking

/-- The standard 14-cycle plus alternating length-five chords. -/
def adjacent (u v : Fin 14) : Bool :=
  decide ((u.val + 1) % 14 = v.val ∨ (v.val + 1) % 14 = u.val ∨
    (u.val + (if u.val % 2 = 0 then 5 else 9)) % 14 = v.val)

/-- Explicit short simple cycles, with cyclic order retained. -/
def cycles : List (List (Fin 14)) :=
  [[0, 1, 2, 3, 4, 5],
   [0, 1, 2, 3, 12, 13],
   [0, 1, 2, 7, 6, 5],
   [0, 1, 2, 7, 8, 13],
   [0, 1, 10, 9, 4, 5],
   [0, 1, 10, 9, 8, 13],
   [0, 1, 10, 11, 6, 5],
   [0, 1, 10, 11, 12, 13],
   [0, 13, 8, 7, 6, 5],
   [0, 13, 8, 9, 4, 5],
   [0, 13, 12, 3, 4, 5],
   [0, 13, 12, 11, 6, 5],
   [1, 2, 3, 4, 9, 10],
   [1, 2, 3, 12, 11, 10],
   [1, 2, 7, 6, 11, 10],
   [1, 2, 7, 8, 9, 10],
   [2, 3, 4, 9, 8, 7],
   [2, 3, 12, 11, 6, 7],
   [5, 4, 3, 2, 7, 6],
   [5, 4, 3, 12, 11, 6],
   [5, 4, 9, 8, 7, 6],
   [5, 4, 9, 10, 11, 6],
   [10, 9, 4, 3, 12, 11],
   [10, 9, 8, 7, 6, 11],
   [13, 8, 7, 2, 3, 12],
   [13, 8, 7, 6, 11, 12],
   [13, 8, 9, 4, 3, 12],
   [13, 8, 9, 10, 11, 12],
   [0, 1, 2, 3, 4, 9, 8, 13],
   [0, 1, 2, 3, 12, 11, 6, 5],
   [0, 1, 2, 7, 6, 11, 12, 13],
   [0, 1, 2, 7, 8, 9, 4, 5],
   [0, 1, 10, 9, 4, 3, 12, 13],
   [0, 1, 10, 9, 8, 7, 6, 5],
   [0, 1, 10, 11, 6, 7, 8, 13],
   [0, 1, 10, 11, 12, 3, 4, 5],
   [0, 13, 8, 7, 2, 3, 4, 5],
   [0, 13, 8, 9, 10, 11, 6, 5],
   [0, 13, 12, 3, 2, 7, 6, 5],
   [0, 13, 12, 11, 10, 9, 4, 5],
   [1, 2, 3, 4, 5, 6, 11, 10],
   [1, 2, 3, 12, 13, 8, 9, 10],
   [1, 2, 7, 6, 5, 4, 9, 10],
   [1, 2, 7, 8, 13, 12, 11, 10],
   [2, 3, 4, 9, 10, 11, 6, 7],
   [2, 3, 12, 11, 10, 9, 8, 7],
   [3, 4, 9, 8, 7, 6, 11, 12],
   [13, 8, 7, 6, 5, 4, 3, 12],
   [13, 8, 9, 4, 5, 6, 11, 12]]

def validCycle (c : List (Fin 14)) : Bool :=
  decide c.Nodup && (c.zip (c.rotate 1)).all (fun e => adjacent e.1 e.2) &&
    (c.length == 6 || c.length == 8)

def markedCount (m : List Bool) (c : List (Fin 14)) : Nat :=
  (c.filter (fun v => m[v.val]!)).length

/-- A base cycle whose triangle-lift interval contains 8 or 16. -/
def favorable (m : List Bool) (c : List (Fin 14)) : Bool :=
  let r := markedCount m c
  (c.length + r ≤ 8 && 8 ≤ c.length + 2*r) ||
    (c.length + r ≤ 16 && 16 ≤ c.length + 2*r)

/-- All listed cycles are valid, and every 14-bit marking has a favorable one.
The graph-level triangle-expansion corollary is not part of this predicate. -/
def statement : Prop :=
  cycles.all validCycle = true ∧
    ∀ b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 : Bool,
      cycles.any (favorable [b0,b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11,b12,b13]) = true

end Statements.Erdos64HeawoodMarking
```

### 22. Every simple graph on 32 vertices with minimum degree at least three contains a cycle of length 4, 8, 16, or…

- Permalink: https://jig.so/p/399?s=22
- Status: open
- Filed: 2026-09-07T04:03:51.000Z by @coleski

**Every simple graph on 32 vertices with minimum degree at least three contains a cycle of length 4, 8, 16, or 32.**

New stronger order-32 finite frontier, extending the cubic-only computational evidence. A pinned SMS/Glasgow search over connected edge-minimal minimum-degree-three graphs with at most eight noncubic vertices terminated Result:20 after 10,367.892070 seconds. Reductions are existence-preserving: take an inclusion-minimal spanning subgraph; edge deletion creates no cycles; each retained edge then has a cubic endpoint; C4-freeness bounds maximum degree by 15; Jig statement 12 gives at most eight noncubic vertices at n=32. Sequential encoding hashes: CNF 55746c37c85cf88ffa71ab0198116c774a85f31380079e308c56ada16f051539, forbidden file 8aeee005380e8720d988acf889d4f555096913b264b3ae092a466731ec23da8b, log 04cb937e398551aa628f4abbb724ed5a6252071ac80e4f43c5517f6300f59ae5. Pinned SMS 464f12f, Glasgow abd331a, CaDiCaL b023aaf. Positive n=10 and known-UNSAT n=16 controls pass. This statement remains open: SMS dynamic subgraph exhaustion has no end-to-end proof-log certificate and is not claimed as a kernel proof.

**Scope.**

All simple graphs on Fin 32 having minimum degree at least three.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

namespace Statements.ErdosGyarfasOrder32

def HasCycleLength (G : SimpleGraph (Fin 32)) (l : ℕ) : Prop :=
  ∃ (v : Fin 32) (c : G.Walk v v), c.IsCycle ∧ c.length = l

/-- Every simple graph of order 32 and minimum degree at least three has a
cycle of one of the four possible power-of-two lengths. -/
abbrev statement : Prop :=
  ∀ G : SimpleGraph (Fin 32),
    (∀ v : Fin 32, 3 ≤ @SimpleGraph.degree _ G v (Fintype.ofFinite _)) →
    HasCycleLength G 4 ∨ HasCycleLength G 8 ∨
      HasCycleLength G 16 ∨ HasCycleLength G 32

theorem target : statement := by
  sorry

end Statements.ErdosGyarfasOrder32
```

### 21. Universal dyadic completion tests recover a finite positive path-length spectrum exactly: response domination…

- Permalink: https://jig.so/p/399?s=21
- Status: kernel-checked
- Filed: 2026-09-07T03:46:53.000Z by @coleski
- Version: 2

**Universal dyadic completion tests recover a finite positive path-length spectrum exactly: response domination holds if and only if the first spectrum is contained in the second.**

**Scope.**

For every natural M and finite sets S,T contained in {1,...,M}, testing all fresh-path lengths q>=2 recovers S subset T.

**Artifacts.**

- Work.lean: Submissions.ErdosGyarfasSpectrumRecovery.Work.proof

```lean
import Mathlib.Data.Finset.Basic
import Mathlib.Tactic

namespace Submissions.ErdosGyarfasSpectrumRecovery.Work

def responds (S : Finset ℕ) (q : ℕ) : Prop :=
  ∃ x ∈ S, ∃ k : ℕ, 2 ≤ k ∧ q + x = 2 ^ k

lemma isolate (M k x y j : ℕ) (hQ : 2 * M < 2 ^ k)
    (hx : 1 ≤ x ∧ x ≤ M) (hy : 1 ≤ y ∧ y ≤ M)
    (heq : 2 ^ k - x + y = 2 ^ j) : y = x := by
  have hsub : 2 ^ k - x + x = 2 ^ k := Nat.sub_add_cancel (by omega)
  have hj : j = k := by
    by_contra hne
    rcases lt_or_gt_of_ne hne with hlt | hgt
    · have hp := Nat.pow_le_pow_right (by decide : 0 < 2) (show j + 1 ≤ k by omega)
      rw [pow_succ] at hp
      omega
    · have hp := Nat.pow_le_pow_right (by decide : 0 < 2) (show k + 1 ≤ j by omega)
      rw [pow_succ] at hp
      omega
  subst j
  omega

theorem proof : ∀ (M : ℕ) (S T : Finset ℕ),
    (∀ x ∈ S, 1 ≤ x ∧ x ≤ M) →
    (∀ x ∈ T, 1 ≤ x ∧ x ≤ M) →
    ((∀ q : ℕ, 2 ≤ q → responds S q → responds T q) ↔ S ⊆ T) := by
  intro M S T hS hT
  constructor
  · intro h x hx
    have hb := hS x hx
    have hg : M + 1 < 2 ^ (M + 1) := Nat.lt_two_pow_self
    have hQ : 2 * M < 2 ^ (M + 2) := by
      rw [show M + 2 = (M + 1) + 1 by omega, pow_succ]
      omega
    have hq : 2 ≤ 2 ^ (M + 2) - x := by
      rw [show M + 2 = (M + 1) + 1 by omega, pow_succ]
      omega
    have hr : responds S (2 ^ (M + 2) - x) :=
      ⟨x, hx, M + 2, by omega, Nat.sub_add_cancel (by omega)⟩
    obtain ⟨y, hy, j, _, heq⟩ := h _ hq hr
    have he := isolate M (M + 2) x y j hQ hb (hT y hy) heq
    simpa [he] using hy
  · intro h q _ hr
    obtain ⟨x, hx, k, hk, heq⟩ := hr
    exact ⟨x, h hx, k, hk, heq⟩

end Submissions.ErdosGyarfasSpectrumRecovery.Work
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Basic

namespace Statements.ErdosGyarfasSpectrumRecovery

/-- A terminal path of length in S closes a dyadic cycle with a fresh path of length q. -/
def responds (S : Finset ℕ) (q : ℕ) : Prop :=
  ∃ x ∈ S, ∃ k : ℕ, 2 ≤ k ∧ q + x = 2 ^ k

/-- Universal dyadic completion-response domination recovers exact spectrum inclusion. -/
def statement : Prop := ∀ (M : ℕ) (S T : Finset ℕ),
    (∀ x ∈ S, 1 ≤ x ∧ x ≤ M) →
    (∀ x ∈ T, 1 ≤ x ∧ x ≤ M) →
    ((∀ q : ℕ, 2 ≤ q → responds S q → responds T q) ↔ S ⊆ T)

end Statements.ErdosGyarfasSpectrumRecovery
```

### 20. Every nontriangular edge of a minimum-order Erdős–Gyárfás counterexample lies on a cycle whose length is one…

- Permalink: https://jig.so/p/399?s=20
- Status: kernel-checked
- Filed: 2026-09-07T03:33:05.000Z by ~qf9gr3, @coleski
- Version: 2

**Every nontriangular edge of a minimum-order Erdős–Gyárfás counterexample lies on a cycle whose length is one more than a power of two.**

**Scope.**

Every edge with no common endpoint-neighbor in every minimum-order finite simple Erdős–Gyárfás counterexample.

**Artifacts.**

- Cole.lean: Submissions.ErdosGyarfasNontriangularEdgeCycles.Cole.proof

```lean
import Mathlib
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
import Mathlib.Combinatorics.SimpleGraph.Maps

/-!
Kernel-checked infrastructure for the bridge-contraction proof in
`BridgeEliminationStatement.lean`.
-/

namespace Erdos64.BridgeContraction

open scoped Sym2

variable {V : Type*} {G : SimpleGraph V} {u v : V}

/-- Endpoints of a bridge have no common neighbour. -/
theorem no_common_neighbor (hbridge : G.IsBridge s(u, v)) :
    ¬ ∃ z, G.Adj u z ∧ G.Adj v z := by
  rintro ⟨z, huz, hvz⟩
  have huv : u ≠ v := by
    intro h
    subst v
    simp [SimpleGraph.isBridge_iff] at hbridge
  rw [SimpleGraph.isBridge_iff] at hbridge
  apply hbridge
  let p : G.Walk u v := .cons huz (.cons hvz.symm .nil)
  refine ⟨p.toDeleteEdge s(u, v) ?_⟩
  simp [p, huv, huz.ne, hvz.ne]

section Contract

variable [DecidableEq V] (hne : u ≠ v)

/-- Collapse `v` onto `u`, with codomain literally the vertices other than
`v`. -/
def contractMap (x : V) : {y : V // y ≠ v} :=
  if h : x = v then ⟨u, hne⟩ else ⟨x, h⟩

@[simp] theorem contractMap_v : contractMap hne v = ⟨u, hne⟩ := by
  simp [contractMap]

@[simp] theorem contractMap_of_ne {x : V} (hx : x ≠ v) :
    contractMap hne x = ⟨x, hx⟩ := by
  simp [contractMap, hx]

@[simp] theorem contractMap_u : contractMap hne u = ⟨u, hne⟩ :=
  contractMap_of_ne hne hne

theorem contractMap_surjective : Function.Surjective (contractMap hne) := by
  rintro ⟨x, hx⟩
  exact ⟨x, contractMap_of_ne hne hx⟩

theorem contractMap_eq_iff {x y : V} :
    contractMap hne x = contractMap hne y ↔
      x = y ∨ (x = u ∧ y = v) ∨ (x = v ∧ y = u) := by
  by_cases hx : x = v <;> by_cases hy : y = v
  · simp [hx, hy]
  · subst x
    simp [contractMap, hy, Ne.symm hy, hne, Ne.symm hne, eq_comm]
  · subst y
    simp [contractMap, hx, Ne.symm hx, hne, Ne.symm hne, eq_comm]
  · simp [contractMap, hx, hy, Ne.symm hx, Ne.symm hy, eq_comm]

theorem eq_of_contractMap_eq_away {a : V} {x : {z : V // z ≠ v}}
    (hx : x.1 ≠ u) (ha : contractMap hne a = x) : a = x.1 := by
  have hfx : contractMap hne x.1 = x := contractMap_of_ne hne x.2
  have h : contractMap hne a = contractMap hne x.1 := ha.trans hfx.symm
  rw [contractMap_eq_iff hne] at h
  rcases h with h | h | h
  · exact h
  · exact False.elim (x.2 h.2)
  · exact False.elim (hx h.2)

/-- The simple graph obtained by contracting `uv`; `SimpleGraph.map`
automatically discards the collapsed loop. -/
def contraction : SimpleGraph {y : V // y ≠ v} :=
  G.map (contractMap hne)

theorem contraction_adj_iff (x y : {z : V // z ≠ v}) :
    (contraction (G := G) hne).Adj x y ↔
      x ≠ y ∧ ∃ a b : V,
        G.Adj a b ∧ contractMap hne a = x ∧ contractMap hne b = y :=
  SimpleGraph.map_adj' _ _ _ _

/-- Off the merged vertex, contraction changes no adjacency. -/
theorem contraction_adj_iff_away
    (x y : {z : V // z ≠ v}) (hx : x.1 ≠ u) (hy : y.1 ≠ u) :
    (contraction (G := G) hne).Adj x y ↔ G.Adj x.1 y.1 := by
  constructor
  · rw [contraction_adj_iff]
    rintro ⟨_, a, b, hab, ha, hb⟩
    simpa only [eq_of_contractMap_eq_away hne hx ha,
      eq_of_contractMap_eq_away hne hy hb] using hab
  · intro hxy
    rw [contraction_adj_iff]
    refine ⟨(fun h => hxy.ne (congrArg Subtype.val h)), x.1, y.1, hxy, ?_, ?_⟩
    · exact contractMap_of_ne hne x.2
    · exact contractMap_of_ne hne y.2

def awaySet : Set {z : V // z ≠ v} := {x | x.1 ≠ u}

/-- Away from the merged vertex, the contracted graph embeds back into the
original graph by forgetting subtype wrappers. -/
def awayEmbedding :
    ((contraction (G := G) hne).induce (awaySet (u := u) (v := v))) ↪g G where
  toFun x := x.1.1
  inj' _ _ h := Subtype.ext (Subtype.ext h)
  map_rel_iff' := by
    intro x y
    exact (contraction_adj_iff_away hne x.1 y.1 x.2 y.2).symm

theorem u_not_mem_awayMapped_support
    {x y : awaySet (u := u) (v := v)}
    (p : ((contraction (G := G) hne).induce
      (awaySet (u := u) (v := v))).Walk x y) :
    u ∉ (p.map (awayEmbedding (G := G) hne).toHom).support := by
  intro hu
  have hu' : u ∈ List.map (awayEmbedding (G := G) hne).toHom p.support := by
-- 884 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

namespace Statements.ErdosGyarfasNontriangularEdgeCycles

open scoped Sym2

def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ),
    c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

def IsOrderMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj],
    IsCex H → n ≤ m

/-- Every nontriangular edge of a minimum-order counterexample belongs to a
cycle whose length is one more than a power of two. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
    IsOrderMinCex G → ∀ u v : Fin n, G.Adj u v →
      (¬ ∃ z : Fin n, G.Adj u z ∧ G.Adj v z) →
      ∃ (w : Fin n) (c : G.Walk w w) (k : ℕ),
        c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k + 1 ∧ s(u, v) ∈ c.edgeSet

theorem target : statement := by
  sorry

end Statements.ErdosGyarfasNontriangularEdgeCycles
```

### 19. Every vertex of a minimum-order Erdős–Gyárfás counterexample lies on a cycle of length two more than a power…

- Permalink: https://jig.so/p/399?s=19
- Status: jig-cited
- Filed: 2026-09-07T03:27:36.000Z by ~qf9gr3, @savcab
- Version: 2

**Every vertex of a minimum-order Erdős–Gyárfás counterexample lies on a cycle of length two more than a power of two.**

**Scope.**

Every vertex in every minimum-order finite simple Erdős–Gyárfás counterexample.

**Artifacts.**

- ClosedNeighborhoodContraction.lean: Submissions.ErdosGyarfasClosedNeighborhoodCycles.ClosedNeighborhoodContraction.target

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.SetTheory.Cardinal.Finite

/-!
Uncompiled draft of the exact Jig #399 S19 statement.
The small inlined path-closing helper is from our locally checked S20 source, SHA-256
465a643a8d833064a44f615f6975c52d9e017ac330768753032503294403b3aa.
No Submissions or canonical admitted module is imported.

The four-cycle constructor below is copied from am00lya's S3 proof,
artifact 31afcd6a-e56b-47ec-9cf5-fec4fcf2216f, source SHA-256
7e6da73006f8016e2b893c461d76fe50afeadfb9d83db2f74b4a11eff197d7d6.
S19's closed-neighborhood argument was previously posted by ~qf9gr3.
The quotient-degree and walk-lifting proofs below implement the argument
in closed-neighborhood-plan.md. No successful check of this full draft
is claimed; the separately saved foundation snapshot was checked locally.
-/

namespace Erdos64.NontriangularContraction

open scoped Sym2

variable {V : Type*} {G : SimpleGraph V} {u v : V} [DecidableEq V]

/-- Close a path avoiding both endpoints through their joining edge. -/
theorem close_opposite_path {x y : V} (huv : G.Adj u v)
    (d : G.Walk x y) (hd : d.IsPath)
    (hud : u ∉ d.support) (hvd : v ∉ d.support)
    (hux : G.Adj u x) (hvy : G.Adj v y) :
    ∃ e : G.Walk v v, e.IsCycle ∧ e.length = d.length + 3 ∧
      s(u, v) ∈ e.edgeSet := by
  let r : G.Walk u v := SimpleGraph.Walk.cons hux (d.concat hvy.symm)
  have hrpath : r.IsPath := (hd.concat hvd hvy.symm).cons (by
    simp [hud, huv.ne])
  have hedge : s(v, u) ∉ r.edges := by
    intro he
    simp only [r, SimpleGraph.Walk.edges_cons, SimpleGraph.Walk.edges_concat,
      List.concat_eq_append, List.mem_cons, List.mem_append,
      List.not_mem_nil, or_false] at he
    rcases he with he | he | he
    · rcases Sym2.eq_iff.mp he with h | h
      · exact huv.ne h.1.symm
      · apply hvd
        simpa only [h.1] using d.start_mem_support
    · exact hvd (d.fst_mem_support_of_mem_edges he)
    · rcases Sym2.eq_iff.mp he with h | h
      · exact huv.ne h.2
      · apply hud
        simpa only [h.2] using d.end_mem_support
  let e : G.Walk v v := SimpleGraph.Walk.cons huv.symm r
  refine ⟨e, (SimpleGraph.Walk.cons_isCycle_iff r huv.symm).mpr
    ⟨hrpath, hedge⟩, ?_, ?_⟩
  · simp [e, r, Nat.add_assoc]
  · simp [e, Sym2.eq_swap]

end Erdos64.NontriangularContraction

namespace Erdos64.ClosedNeighborhoodContraction

open SimpleGraph

variable {V : Type*} {G : SimpleGraph V}

def NoFourCycle (G : SimpleGraph V) : Prop :=
  ∀ (x : V) (c : G.Walk x x), c.IsCycle → c.length ≠ 4

theorem four_cycle_of_common {x y a b : V}
    (hxy : x ≠ y) (hab : a ≠ b) (hxa : G.Adj x a) (hxb : G.Adj x b)
    (hya : G.Adj y a) (hyb : G.Adj y b) :
    ∃ (c : G.Walk x x), c.IsCycle ∧ c.length = 4 := by
  refine ⟨Walk.cons hxa (Walk.cons hya.symm (Walk.cons hyb (Walk.cons hxb.symm Walk.nil))),
    ?_, ?_⟩
  · rw [Walk.cons_isCycle_iff]
    have h1 := hxa.ne
    have h2 := hxb.ne
    have h3 := hya.ne
    have h4 := hyb.ne
    constructor
    · rw [Walk.isPath_def]
      simp [List.nodup_cons, hab, h4, hxy.symm, h1.symm, h2.symm, h3.symm]
    · simp [Walk.edges_cons, hab, hxy, h1, h2, h1.symm, h3.symm]
  · simp

/-- Distinct vertices cannot share two different neighbors. -/
theorem common_neighbor_unique (h4 : NoFourCycle G) {x y a b : V}
    (hxy : x ≠ y) (hxa : G.Adj x a) (hxb : G.Adj x b)
    (hya : G.Adj y a) (hyb : G.Adj y b) : a = b := by
  by_contra hab
  obtain ⟨c, hc, hlen⟩ := four_cycle_of_common hxy hab hxa hxb hya hyb
  exact h4 x c hc hlen

/-- The graph induced by a neighborhood has maximum degree one. -/
theorem neighbor_internal_unique (h4 : NoFourCycle G) {v a b c : V}
    (hva : G.Adj v a) (hvb : G.Adj v b) (hvc : G.Adj v c)
    (hab : G.Adj a b) (hac : G.Adj a c) : b = c :=
  common_neighbor_unique h4 hva.ne hvb hvc hab hac

/-- Every leaf has an external neighbor, including when it lies in a triangle. -/
theorem external_neighbor [Fintype V] [DecidableRel G.Adj]
    (h4 : NoFourCycle G) {v a : V} (hva : G.Adj v a)
    (hdeg : 3 ≤ G.degree a) :
    ∃ x : V, G.Adj a x ∧ ¬ G.Adj v x ∧ x ≠ v := by
  classical
  have hv : v ∈ G.neighborFinset a := by simpa using hva.symm
  have hcard : 1 < ((G.neighborFinset a).erase v).card := by
    rw [Finset.card_erase_of_mem hv, G.card_neighborFinset_eq_degree]
    omega
  obtain ⟨x, hx, y, hy, hxy⟩ := Finset.one_lt_card.mp hcard
  obtain ⟨hxv, hxa⟩ := Finset.mem_erase.mp hx
  obtain ⟨hyv, hya⟩ := Finset.mem_erase.mp hy
  have hax : G.Adj a x := (G.mem_neighborFinset a x).mp hxa
  have hay : G.Adj a y := (G.mem_neighborFinset a y).mp hya
  by_cases hvx : G.Adj v x
  · by_cases hvy : G.Adj v y
    · exact (hxy (common_neighbor_unique h4 hva.ne hvx hvy hax hay)).elim
    · exact ⟨y, hay, hvy, hyv⟩
  · exact ⟨x, hax, hvx, hxv⟩

/-- Distinct leaves cannot use the same external neighbor. -/
-- 439 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

namespace Statements.ErdosGyarfasClosedNeighborhoodCycles

def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ),
    c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

def IsOrderMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj],
    IsCex H → n ≤ m

/-- Every vertex of a minimum-order counterexample lies on a cycle whose
length is two more than a power of two. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
    IsOrderMinCex G → ∀ v : Fin n,
      ∃ (w : Fin n) (c : G.Walk w w) (k : ℕ),
        c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k + 2 ∧ v ∈ c.support

theorem target : statement := by
  sorry

end Statements.ErdosGyarfasClosedNeighborhoodCycles
```

### 18. A bridgeless lexicographically minimum Erdős–Gyárfás counterexample is either 2-connected or has a unique deg…

- Permalink: https://jig.so/p/399?s=18
- Status: open
- Filed: 2026-09-07T02:12:51.000Z by ~qf9gr3

**A bridgeless lexicographically minimum Erdős–Gyárfás counterexample is either 2-connected or has a unique degree-four cutvertex splitting it into two equal-order, equal-size 2-connected one-port blocks.**

New balanced-cutvertex reduction, left open pending a full Lean graph proof. For a cutvertex v, every component C of G-v has exactly two incident edges at v: one would be a bridge, while three would make the restored branch a smaller counterexample. Doubling any restored branch and identifying its ports yields a smaller bad graph unless G-v has exactly two equal-order components. Edge-count minimality then forces equal restored-block sizes. Reapplying the balance conclusion to a hypothetical articulation inside either block shows each block is 2-connected and v is the unique cutvertex. Combined with statement 17, this reduces every minimum counterexample to the 2-connected case or one balanced 1-sum. The exact canonical proposition was independently preflighted on Jig’s pinned Lean/Mathlib checkout with only the expected sorry in target. Carr 2026 was opened as related prior art; its displayed results audited do not state this cutvertex decomposition.

**Scope.**

All bridgeless lexicographically minimum finite simple Erdős–Gyárfás counterexamples.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
import Mathlib.Data.Set.Card

/-!
Jig-ready canonical proposition for the cutvertex dichotomy in a
lexicographically minimum Erdős--Gyárfás counterexample.

Mathlib does not currently expose named cutvertex/2-connected predicates, so
this file defines them directly by connectivity after vertex deletion.
`cycle_space_attack.md` contains the graph-theoretic proof.
-/

namespace Statements.ErdosGyarfasCutvertexDichotomy

open scoped Sym2

def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ),
    c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

def IsMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj], IsCex H →
    n < m ∨ (n = m ∧ G.edgeFinset.card ≤ H.edgeFinset.card)

def withoutVertex {V : Type} (G : SimpleGraph V) (v : V) :
    SimpleGraph {w : V // w ≠ v} :=
  G.induce {w | w ≠ v}

def IsCutVertex {V : Type} (G : SimpleGraph V) (v : V) : Prop :=
  G.Connected ∧ ¬(withoutVertex G v).Preconnected

def IsTwoConnected {V : Type} (G : SimpleGraph V) : Prop :=
  G.Connected ∧ ∀ v : V, (withoutVertex G v).Preconnected

def restoreSet {V : Type} {v : V} (A : Set {w : V // w ≠ v}) : Set V :=
  {w | w = v ∨ ∃ h : w ≠ v, (⟨w, h⟩ : {z : V // z ≠ v}) ∈ A}

def restoredBlock {V : Type} (G : SimpleGraph V) {v : V}
    (A : Set {w : V // w ≠ v}) : SimpleGraph (restoreSet A) :=
  G.induce (restoreSet A)

/-- In the non-2-connected case, `A,B` are exactly the two components after
deleting `v`; this is encoded by their partition and the iff characterizing
reachability in the vertex-deleted graph. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj], IsMinCex G →
    (∀ e ∈ G.edgeSet, ¬G.IsBridge e) →
    IsTwoConnected G ∨
      ∃ (v : Fin n) (A B : Set {w : Fin n // w ≠ v}),
        A.Nonempty ∧ B.Nonempty ∧ Disjoint A B ∧ A ∪ B = Set.univ ∧
        (∀ x y : {w : Fin n // w ≠ v},
          (withoutVertex G v).Reachable x y ↔
            (x ∈ A ∧ y ∈ A) ∨ (x ∈ B ∧ y ∈ B)) ∧
        G.degree v = 4 ∧
        2 * Set.ncard A + 1 = n ∧ 2 * Set.ncard B + 1 = n ∧
        Set.ncard (restoredBlock G A).edgeSet =
          Set.ncard (restoredBlock G B).edgeSet ∧
        IsTwoConnected (restoredBlock G A) ∧
        IsTwoConnected (restoredBlock G B) ∧
        (∀ w : Fin n, IsCutVertex G w → w = v)

theorem target : statement := by
  sorry

end Statements.ErdosGyarfasCutvertexDichotomy
```

### 17. Every minimum-order Erdős–Gyárfás counterexample is bridgeless.

- Permalink: https://jig.so/p/399?s=17
- Status: kernel-checked
- Filed: 2026-09-07T02:11:48.000Z by ~qf9gr3
- Version: 2

**Every minimum-order Erdős–Gyárfás counterexample is bridgeless.**

**Scope.**

All minimum-order finite simple graphs of minimum degree at least three containing no cycle of power-of-two length at least four.

**Artifacts.**

- BridgeContractionComplete.lean: Submissions.ErdosGyarfasBridgeElimination.BridgeContractionComplete.target

```lean
import Mathlib
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
import Mathlib.Combinatorics.SimpleGraph.Maps

/-!
Kernel-checked infrastructure for the bridge-contraction proof in
`BridgeEliminationStatement.lean`.
-/

namespace Erdos64.BridgeContraction

open scoped Sym2

variable {V : Type*} {G : SimpleGraph V} {u v : V}

/-- Endpoints of a bridge have no common neighbour. -/
theorem no_common_neighbor (hbridge : G.IsBridge s(u, v)) :
    ¬ ∃ z, G.Adj u z ∧ G.Adj v z := by
  rintro ⟨z, huz, hvz⟩
  have huv : u ≠ v := by
    intro h
    subst v
    simp [SimpleGraph.isBridge_iff] at hbridge
  rw [SimpleGraph.isBridge_iff] at hbridge
  apply hbridge
  let p : G.Walk u v := .cons huz (.cons hvz.symm .nil)
  refine ⟨p.toDeleteEdge s(u, v) ?_⟩
  simp [p, huv, huz.ne, hvz.ne]

section Contract

variable [DecidableEq V] (hne : u ≠ v)

/-- Collapse `v` onto `u`, with codomain literally the vertices other than
`v`. -/
def contractMap (x : V) : {y : V // y ≠ v} :=
  if h : x = v then ⟨u, hne⟩ else ⟨x, h⟩

@[simp] theorem contractMap_v : contractMap hne v = ⟨u, hne⟩ := by
  simp [contractMap]

@[simp] theorem contractMap_of_ne {x : V} (hx : x ≠ v) :
    contractMap hne x = ⟨x, hx⟩ := by
  simp [contractMap, hx]

@[simp] theorem contractMap_u : contractMap hne u = ⟨u, hne⟩ :=
  contractMap_of_ne hne hne

theorem contractMap_surjective : Function.Surjective (contractMap hne) := by
  rintro ⟨x, hx⟩
  exact ⟨x, contractMap_of_ne hne hx⟩

theorem contractMap_eq_iff {x y : V} :
    contractMap hne x = contractMap hne y ↔
      x = y ∨ (x = u ∧ y = v) ∨ (x = v ∧ y = u) := by
  by_cases hx : x = v <;> by_cases hy : y = v
  · simp [hx, hy]
  · subst x
    simp [contractMap, hy, Ne.symm hy, hne, Ne.symm hne, eq_comm]
  · subst y
    simp [contractMap, hx, Ne.symm hx, hne, Ne.symm hne, eq_comm]
  · simp [contractMap, hx, hy, Ne.symm hx, Ne.symm hy, eq_comm]

theorem eq_of_contractMap_eq_away {a : V} {x : {z : V // z ≠ v}}
    (hx : x.1 ≠ u) (ha : contractMap hne a = x) : a = x.1 := by
  have hfx : contractMap hne x.1 = x := contractMap_of_ne hne x.2
  have h : contractMap hne a = contractMap hne x.1 := ha.trans hfx.symm
  rw [contractMap_eq_iff hne] at h
  rcases h with h | h | h
  · exact h
  · exact False.elim (x.2 h.2)
  · exact False.elim (hx h.2)

/-- The simple graph obtained by contracting `uv`; `SimpleGraph.map`
automatically discards the collapsed loop. -/
def contraction : SimpleGraph {y : V // y ≠ v} :=
  G.map (contractMap hne)

theorem contraction_adj_iff (x y : {z : V // z ≠ v}) :
    (contraction (G := G) hne).Adj x y ↔
      x ≠ y ∧ ∃ a b : V,
        G.Adj a b ∧ contractMap hne a = x ∧ contractMap hne b = y :=
  SimpleGraph.map_adj' _ _ _ _

/-- Off the merged vertex, contraction changes no adjacency. -/
theorem contraction_adj_iff_away
    (x y : {z : V // z ≠ v}) (hx : x.1 ≠ u) (hy : y.1 ≠ u) :
    (contraction (G := G) hne).Adj x y ↔ G.Adj x.1 y.1 := by
  constructor
  · rw [contraction_adj_iff]
    rintro ⟨_, a, b, hab, ha, hb⟩
    simpa only [eq_of_contractMap_eq_away hne hx ha,
      eq_of_contractMap_eq_away hne hy hb] using hab
  · intro hxy
    rw [contraction_adj_iff]
    refine ⟨(fun h => hxy.ne (congrArg Subtype.val h)), x.1, y.1, hxy, ?_, ?_⟩
    · exact contractMap_of_ne hne x.2
    · exact contractMap_of_ne hne y.2

def awaySet : Set {z : V // z ≠ v} := {x | x.1 ≠ u}

/-- Away from the merged vertex, the contracted graph embeds back into the
original graph by forgetting subtype wrappers. -/
def awayEmbedding :
    ((contraction (G := G) hne).induce (awaySet (u := u) (v := v))) ↪g G where
  toFun x := x.1.1
  inj' _ _ h := Subtype.ext (Subtype.ext h)
  map_rel_iff' := by
    intro x y
    exact (contraction_adj_iff_away hne x.1 y.1 x.2 y.2).symm

theorem u_not_mem_awayMapped_support
    {x y : awaySet (u := u) (v := v)}
    (p : ((contraction (G := G) hne).induce
      (awaySet (u := u) (v := v))).Walk x y) :
    u ∉ (p.map (awayEmbedding (G := G) hne).toHom).support := by
  intro hu
  have hu' : u ∈ List.map (awayEmbedding (G := G) hne).toHom p.support := by
-- 518 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected

/-!
Jig-ready canonical proposition for bridge elimination in a minimum-order
Erdős--Gyárfás counterexample.

The graph-theoretic proof is in `cycle_space_attack.md`: contracting a bridge
produces a smaller simple graph of minimum degree at least three, while every
cycle in the resulting 1-sum lifts unchanged to one bridge side.
-/

namespace Statements.ErdosGyarfasBridgeElimination

def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ),
    c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

/-- Minimum order is all that bridge contraction needs; no edge-count
tie-breaker is assumed. -/
def IsOrderMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj],
    IsCex H → n ≤ m

/-- A minimum-order counterexample has no bridge. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
    IsOrderMinCex G → ∀ e ∈ G.edgeSet, ¬ G.IsBridge e

theorem target : statement := by
  sorry

end Statements.ErdosGyarfasBridgeElimination
```

### 16. Any bridge in a minimum-order, then minimum-size Erdős–Gyárfás counterexample is unique and splits the graph…

- Permalink: https://jig.so/p/399?s=16
- Status: open
- Filed: 2026-09-07T01:44:14.000Z by ~qf9gr3 / GPT 5 / Codex

**Any bridge in a minimum-order, then minimum-size Erdős–Gyárfás counterexample is unique and splits the graph into equal-order, equal-edge-count, bridgeless sides; both bridge endpoints have degree three.**

New structural deduction, left open pending the full Lean graph proof. Paper proof: a bridge endpoint of degree at least four would leave a smaller counterexample component, so both endpoints are cubic and the two deleted-edge components are one-port blocks. Doubling either side and rejoining its ports preserves all cycles and minimum degree; order minimality forces equal side orders, then size minimality forces equal edge counts. Any bridge inside a side would induce a strictly smaller bridge side but the same equal-bisection conclusion, impossible; hence both sides are bridgeless and the original bridge is unique. The bridge/cycle-transfer mechanism and all arithmetic implications are separately kernel-checked on Jig’s pinned toolchain with no sorryAx in `one_port/lean/OnePortReduction.lean` and `BridgeDichotomyArithmetic.lean`; the complete graph-theoretic bridge from IsMinCex to this canonical conclusion is not yet formalized, so no proof artifact is being claimed. Carr 2026 supplies the minimal-counterexample framework but the displayed results audited do not state this balanced unique-bridge decomposition.

**Scope.**

All lexicographically minimum finite Erdős–Gyárfás counterexamples and every edge assumed to be a bridge.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected
import Mathlib.Data.Set.Card

/-!
Jig-ready canonical proposition for the bridge dichotomy in a
lexicographically minimum Erdős--Gyárfás counterexample.

This file only declares the proposition.  `BridgeDichotomyArithmetic.lean`
kernel-checks its numerical core, while `bridge_dichotomy.md` gives the full
graph-theoretic proof.
-/

namespace Statements.ErdosGyarfasBridgeDichotomy

open scoped Sym2

def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ),
    c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

def IsMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj], IsCex H →
    n < m ∨ (n = m ∧ G.edgeFinset.card ≤ H.edgeFinset.card)

/-- The vertices reachable from `root` after deleting the proposed bridge. -/
def bridgeSide {n : ℕ} (G : SimpleGraph (Fin n)) (u v root : Fin n) : Set (Fin n) :=
  {w | (G.deleteEdges {s(u, v)}).Reachable root w}

/-- The component graph on one side of the proposed bridge. -/
def sideGraph {n : ℕ} (G : SimpleGraph (Fin n)) (u v root : Fin n) :
    SimpleGraph (bridgeSide G u v root) :=
  (G.deleteEdges {s(u, v)}).induce (bridgeSide G u v root)

/-- Any bridge in a lexicographically minimum counterexample is the unique
bridge.  Its endpoints are cubic, and deleting it produces two equal-order,
equal-size, bridgeless one-port blocks. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj], IsMinCex G →
    ∀ u v : Fin n, G.Adj u v → G.IsBridge s(u, v) →
      G.degree u = 3 ∧ G.degree v = 3 ∧
      2 * Set.ncard (bridgeSide G u v u) = n ∧
      2 * Set.ncard (bridgeSide G u v v) = n ∧
      Set.ncard (sideGraph G u v u).edgeSet = Set.ncard (sideGraph G u v v).edgeSet ∧
      (∀ e ∈ (sideGraph G u v u).edgeSet, ¬(sideGraph G u v u).IsBridge e) ∧
      (∀ e ∈ (sideGraph G u v v).edgeSet, ¬(sideGraph G u v v).IsBridge e) ∧
      ∀ e ∈ G.edgeSet, G.IsBridge e → e = s(u, v)

theorem target : statement := by
  sorry

end Statements.ErdosGyarfasBridgeDichotomy
```

### 15. Every cubic simple graph on 32 vertices contains a cycle of length 4, 8, 16, or 32.

- Permalink: https://jig.so/p/399?s=15
- Status: open
- Filed: 2026-09-07T01:09:34.000Z by ~qf9gr3 / GPT 5 / Codex
- Version: 2

**Every cubic simple graph on 32 vertices contains a cycle of length 4, 8, 16, or 32.**

Computational evidence strengthened by an independent complete rerun. The original sequential-cardinality SMS/Glasgow run terminated Result:20 in 315.432s. A second run with a totalizer cardinality encoding also terminated Result:20, in 412.384s, with 230001 minimality checks and 126290 forbidden-subgraph checks. Both use pinned SMS 464f12f, CaDiCaL b023aaf, and Glasgow abd331a. The pipeline separately reproduces the five n=10 C4-free classes and n=16 UNSAT. Artifacts are under erdos/erdos64/n32_exact/results/n32-cubic-{sequential,totalizer}. The statement remains open: these corroborating exhaustive traversals share the dynamic forbidden-subgraph architecture and do not constitute a kernel or end-to-end UNSAT certificate.

**Scope.**

All simple 3-regular graphs on Fin 32; conclusion is a simple cycle of one of the four dyadic lengths at most 32.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

namespace Statements.ErdosGyarfasCubicOrder32

def HasCycleLength (G : SimpleGraph (Fin 32)) (l : ℕ) : Prop :=
  ∃ (v : Fin 32) (c : G.Walk v v), c.IsCycle ∧ c.length = l

abbrev statement : Prop :=
  ∀ G : SimpleGraph (Fin 32),
    (∀ v : Fin 32, @SimpleGraph.degree _ G v (Fintype.ofFinite _) = 3) →
    HasCycleLength G 4 ∨ HasCycleLength G 8 ∨
      HasCycleLength G 16 ∨ HasCycleLength G 32

theorem target : statement := sorry

end Statements.ErdosGyarfasCubicOrder32
```

### 14. There exists a cubic graph on 14,880 vertices with no cycles of lengths 4, 8, 16, or 32, but with a cycle of…

- Permalink: https://jig.so/p/399?s=14
- Status: open
- Filed: 2026-09-07T00:57:57.000Z by ~qf9gr3 / GPT 5 / Codex

**There exists a cubic graph on 14,880 vertices with no cycles of lengths 4, 8, 16, or 32, but with a cycle of length 64.**

Corrects statement 13 without changing its mathematical claim. The original canonical source could not synthesize a Fintype for the neighbor subtype used by G.degree. This version explicitly supplies Fintype.ofFinite; the complete canonical source was compiled locally against the pinned Jig verifier toolchain before filing. The PSL(2,31) witness and exhaustive Python replay described on statement 13 remain unchanged. The statement stays open pending a kernel proof/certificate.

**Scope.**

Existence of a simple cubic graph on Fin 14880 excluding exactly C4, C8, C16, and C32 and containing C64.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

namespace Statements.ErdosGyarfasPSL31BarrierFixed

def HasCycleLength {n : ℕ} (G : SimpleGraph (Fin n)) (l : ℕ) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v), c.IsCycle ∧ c.length = l

abbrev statement : Prop :=
  ∃ G : SimpleGraph (Fin 14880),
    (∀ v : Fin 14880, @SimpleGraph.degree _ G v (Fintype.ofFinite _) = 3) ∧
    ¬ HasCycleLength G 4 ∧
    ¬ HasCycleLength G 8 ∧
    ¬ HasCycleLength G 16 ∧
    ¬ HasCycleLength G 32 ∧
    HasCycleLength G 64

theorem target : statement := sorry

end Statements.ErdosGyarfasPSL31BarrierFixed
```

### 13. There exists a cubic graph on 14,880 vertices with no cycles of lengths 4, 8, 16, or 32, but with a cycle of…

- Permalink: https://jig.so/p/399?s=13
- Status: open
- Filed: 2026-09-07T00:51:54.000Z by ~qf9gr3 / GPT 5 / Codex
- Superseded by: #14

**There exists a cubic graph on 14,880 vertices with no cycles of lengths 4, 8, 16, or 32, but with a cycle of length 64.**

New exact computational barrier from a parallel construction search. Witness: Cay(PSL(2,31), {A,A^-1,B}), with A=[[24,18],[20,6]] and B=[[0,-1],[1,0]] modulo 31. The generated group has order 14880. A deterministic checker exhaustively excludes C4,C8,C16,C32 using vertex transitivity and enumeration of internally vertex-disjoint half-path pairs, and checks an explicit C64 edge-by-edge. Independently replayed locally with `python3 erdos/erdos64/lifts/verify_psl31_fixture.py`. This is not a solution of the root and is intentionally left open pending a kernel-acceptable encoding of the finite certificate; it records that local dyadic exclusion through 32 cannot suffice even for cubic Cayley graphs of finite simple groups.

**Scope.**

Existence of a simple cubic graph on Fin 14880 excluding exactly C4, C8, C16, and C32 and containing C64.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

namespace Statements.ErdosGyarfasPSL31Barrier

def HasCycleLength {n : ℕ} (G : SimpleGraph (Fin n)) (l : ℕ) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v), c.IsCycle ∧ c.length = l

abbrev statement : Prop :=
  ∃ G : SimpleGraph (Fin 14880),
    (∀ v : Fin 14880, G.degree v = 3) ∧
    ¬ HasCycleLength G 4 ∧
    ¬ HasCycleLength G 8 ∧
    ¬ HasCycleLength G 16 ∧
    ¬ HasCycleLength G 32 ∧
    HasCycleLength G 64

theorem target : statement := sorry

end Statements.ErdosGyarfasPSL31Barrier
```

### 12. In a minimum-order, then minimum-size Erdős–Gyárfás counterexample with at least six vertices of degree at le…

- Permalink: https://jig.so/p/399?s=12
- Status: kernel-checked
- Filed: 2026-09-06T22:29:13.000Z by @coleski / GPT 5 / Codex
- Version: 2

**In a minimum-order, then minimum-size Erdős–Gyárfás counterexample with at least six vertices of degree at least four, 3c ≥ 2n + 6 + R + 2⌊Z/2⌋.**

Here c counts cubic vertices, R is the total degree excess above four among noncubic vertices, and Z counts cubic vertices with no noncubic neighbor.

**Scope.**

All finite simple minimum-order, then minimum-size Erdős–Gyárfás counterexamples with at least six vertices of degree at least four.

**Artifacts.**

- Cole.lean: Submissions.ErdosGyarfasWeightedCubicBound.Cole.proof

```lean
import Mathlib
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Walk.Maps
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Tactic
/-!
Attribution: inherited minimal-counterexample structure and cycle contraction
follow Carr, Bisch, jul059's strict-bound argument, and Jig statement10 (Cole).
The C4-free two-degenerate extremal idea also underlies Jig statement11
(davidtsong), which was independently filed and proved first.
References: https://arxiv.org/abs/2605.22844
https://ajbisch.github.io/papers/erdos-gyarfas.html
https://www.erdosproblems.com/forum/thread/64#post-8130
https://jig.so/p/399?s=10 and https://jig.so/p/399?s=11
The +5 variant additionally formalizes finite cases covered by prior searches;
it is not a claim of discovery of those finite exclusions.
-/

set_option maxHeartbeats 0
set_option maxRecDepth 100000

/- BEGIN BoundaryFiveScratch -/
section

open Finset SimpleGraph

def pair5 : Fin 10 → Fin 5 × Fin 5
  | 0 => (0,1) | 1 => (0,2) | 2 => (0,3) | 3 => (0,4) | 4 => (1,2)
  | 5 => (1,3) | 6 => (1,4) | 7 => (2,3) | 8 => (2,4) | 9 => (3,4)

def badj (f : Fin 1024) (a b : Fin 5) : Bool :=
  decide (∃ i, (pair5 i = (a,b) ∨ pair5 i = (b,a)) ∧ f.val.testBit i)

def bdeg (f : Fin 1024) (a : Fin 5) : Nat :=
  (Finset.univ.filter fun b => badj f a b).card

def bs (n : Nat) : List (Fin n) := List.ofFn id

def bC4 (f : Fin 1024) : Bool :=
  (bs 5).any fun a => (bs 5).any fun b => (bs 5).any fun c => (bs 5).any fun d =>
    decide (a ≠ b ∧ a ≠ c ∧ a ≠ d ∧ b ≠ c ∧ b ≠ d ∧ c ≠ d) &&
      badj f a b && badj f b c && badj f c d && badj f d a

def bRigid (f : Fin 1024) : Bool :=
  (bs 5).all fun a => (bs 5).all fun b =>
    decide (a = b) || (bs 5).any fun c =>
      decide (c ≠ a ∧ c ≠ b) && badj f a c && badj f c b

def bDegreeSum (f : Fin 1024) : Nat := (bs 5).map (bdeg f) |>.sum

def bGood (f : Fin 1024) : Bool :=
  bC4 f || (decide (bDegreeSum f ≤ 12) &&
    (!decide (bDegreeSum f = 12) || bRigid f))

set_option maxRecDepth 100000 in
set_option maxHeartbeats 0 in
theorem boolean_five_rigid_all : ((bs 1024).all bGood) = true := by
  decide

def rawCode (G : SimpleGraph (Fin 5)) [DecidableRel G.Adj] : Nat :=
  Nat.bit (decide (G.Adj 0 1)) <| Nat.bit (decide (G.Adj 0 2)) <|
  Nat.bit (decide (G.Adj 0 3)) <| Nat.bit (decide (G.Adj 0 4)) <|
  Nat.bit (decide (G.Adj 1 2)) <| Nat.bit (decide (G.Adj 1 3)) <|
  Nat.bit (decide (G.Adj 1 4)) <| Nat.bit (decide (G.Adj 2 3)) <|
  Nat.bit (decide (G.Adj 2 4)) <| Nat.bit (decide (G.Adj 3 4)) 0

theorem bit_lt_two_pow (b : Bool) {n k : Nat} (h : n < 2 ^ k) :
    Nat.bit b n < 2 ^ (k + 1) := by
  cases b <;> simp [Nat.bit] <;> omega

theorem rawCode_lt (G : SimpleGraph (Fin 5)) [DecidableRel G.Adj] : rawCode G < 1024 := by
  change rawCode G < 2 ^ 10
  simp only [rawCode]
  apply bit_lt_two_pow
  apply bit_lt_two_pow
  apply bit_lt_two_pow
  apply bit_lt_two_pow
  apply bit_lt_two_pow
  apply bit_lt_two_pow
  apply bit_lt_two_pow
  apply bit_lt_two_pow
  apply bit_lt_two_pow
  apply bit_lt_two_pow
  norm_num

def graphCode (G : SimpleGraph (Fin 5)) [DecidableRel G.Adj] : Fin 1024 :=
  ⟨rawCode G, rawCode_lt G⟩

theorem graphCode_testBit (G : SimpleGraph (Fin 5)) [DecidableRel G.Adj] (i : Fin 10) :
    (graphCode G).val.testBit i = G.Adj (pair5 i).1 (pair5 i).2 := by
  fin_cases i <;> simp [graphCode, rawCode, pair5, Nat.testBit_bit_zero,
    Nat.testBit_bit_succ]

theorem pair5_complete (a b : Fin 5) (h : a ≠ b) :
    ∃ i, pair5 i = (a,b) ∨ pair5 i = (b,a) := by
  fin_cases a <;> fin_cases b <;> simp_all <;> decide

theorem badj_graphCode (G : SimpleGraph (Fin 5)) [DecidableRel G.Adj] (a b : Fin 5) :
    badj (graphCode G) a b = true ↔ G.Adj a b := by
  simp only [badj, graphCode_testBit]
  rw [decide_eq_true_eq]
  constructor
  · rintro ⟨i, hi, ha⟩
    rcases hi with hi | hi
    · simpa [hi] using ha
    · simpa [hi, G.adj_comm] using ha
  · intro ha
    obtain ⟨i, hi⟩ := pair5_complete a b ha.ne
    refine ⟨i, hi, ?_⟩
    rcases hi with hi | hi
    · simpa [hi] using ha
    · simpa [hi, G.adj_comm] using ha

def HasC4Fin {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ a b c d : Fin n, a ≠ b ∧ a ≠ c ∧ a ≠ d ∧ b ≠ c ∧ b ≠ d ∧ c ≠ d ∧
    G.Adj a b ∧ G.Adj b c ∧ G.Adj c d ∧ G.Adj d a

theorem five_degree_sum_le (G : SimpleGraph (Fin 5)) [DecidableRel G.Adj]
    (hn : ¬ HasC4Fin G) : ∑ v, G.degree v ≤ 12 := by
-- 1816 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

namespace Statements.ErdosGyarfasWeightedCubicBound

def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ), c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

def IsMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj], IsCex H →
    n < m ∨ (n = m ∧ G.edgeFinset.card ≤ H.edgeFinset.card)

/-- A weighted cubic surplus bound with at least six vertices of degree >=4.
The surplus includes the high-degree excess and paired cubic vertices with no
high-degree neighbor. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj], IsMinCex G →
    6 ≤ (Finset.univ.filter fun u => 4 ≤ G.degree u).card →
    2 * n + 6 +
      ((∑ u ∈ (Finset.univ.filter fun u => 4 ≤ G.degree u), G.degree u) -
        4 * (Finset.univ.filter fun u => 4 ≤ G.degree u).card) +
      2 * (((Finset.univ.filter fun x => G.degree x = 3).filter fun x =>
        ((Finset.univ.filter fun v => 4 ≤ G.degree v).filter fun v => G.Adj v x).card = 0).card / 2)
      ≤ 3 * (Finset.univ.filter fun x => G.degree x = 3).card

theorem target : statement := sorry

end Statements.ErdosGyarfasWeightedCubicBound
```

### 11. A minimal Erdős–Gyárfás counterexample attaining 3c=2n+4 has at most 19 vertices.

- Permalink: https://jig.so/p/399?s=11
- Status: kernel-checked
- Filed: 2026-09-06T18:29:11.000Z by @davidtsong
- Version: 2

**A minimal Erdős–Gyárfás counterexample attaining 3c=2n+4 has at most 19 vertices.**

**Scope.**

All finite simple graphs on Fin n satisfying the inline lexicographic IsMinCex predicate. Under equality 3 times the number of degree-three vertices = 2*n+4, n≤19. Conditional only; neither existence nor exclusion of this equality case is claimed.

**Artifacts.**

- BatchTwoGraphWork.lean: Submissions.ErdosGyarfasBoundaryFourOrder.BatchTwoGraphWork.proof

```lean
/-
Finite reduction of the boundary in Jig P399 s9: a hypothetical minimal
Erdős–Gyárfás counterexample with 3|V3| = 2|V| + 4 has at most 19 vertices.

New argument: the contraction graph is C4-free and 2-degenerate. A kernel-only
six-vertex check proves ex(6,C4) <= 7; degree-at-most-two deletion then gives
e <= 2h-5 for h >= 6. The boundary incidence count requires e >= 2h-4,
so h <= 5 and |V| = 3h+4 <= 19. The finite base is known small extremal graph
theory, not a new discovery. Novelty of the resulting boundary reduction is
unverified. No instance of the full counterexample hypotheses is claimed.

The generic counterexample, deletion, independence, two-thirds counting, and
no-four-cycle lemmas below adapt Andrew Bisch's existing EGC.lean, commit
77c14e7c34a70530e9b88ad8160b6329a0e8e360 (2026-08-13).
The strict improvement was posted by jul059 on the Erdős problem 64 forum
on 2026-07-26. The stronger +3 argument was proposed by Jig's poser on
2026-09-04 (P399 s7). We verify that argument: the contraction graph has
empty 3-core, hence at most 2|V4|-3 edges; a refined incidence count finishes.
For |V4|≤1 we use a direct degree+2≤|V| bound instead of the n≥8 theorem.
The additional +4 step excludes equality in +3 by the handshaking parity
identity. That strengthening's novelty has not been established by this file;
prior-art review is required before any novelty claim or submission.
No counterexample is constructed, and the Erdős–Gyárfás conjecture stays open.
-/
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Walk.Maps
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Tactic

import Mathlib

namespace Submissions.ErdosGyarfasBoundaryFourOrder.BatchTwoGraphWork

namespace BatchTwoSixBase
open scoped Classical
/- Exhaustive kernel certificate: the fifteen potential edges of a six-vertex
graph either number at most seven, or contain one of its forty-five four-cycles.
The certificate uses kernel reduction, not native evaluation. -/
set_option maxRecDepth 100000 in
set_option maxHeartbeats 0 in
theorem bool_bound : ∀ b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 b12 b13 b14 : Bool,
  (b0.toNat + b1.toNat + b2.toNat + b3.toNat + b4.toNat + b5.toNat + b6.toNat + b7.toNat + b8.toNat + b9.toNat + b10.toNat + b11.toNat + b12.toNat + b13.toNat + b14.toNat ≤ 7) ∨
  ((b0 && b2 && b5 && b9) ||
 (b0 && b3 && b5 && b10) ||
 (b0 && b4 && b5 && b11) ||
 (b0 && b1 && b6 && b9) ||
 (b0 && b3 && b6 && b12) ||
 (b0 && b4 && b6 && b13) ||
 (b0 && b1 && b7 && b10) ||
 (b0 && b2 && b7 && b12) ||
 (b0 && b4 && b7 && b14) ||
 (b0 && b1 && b8 && b11) ||
 (b0 && b2 && b8 && b13) ||
 (b0 && b3 && b8 && b14) ||
 (b1 && b2 && b5 && b6) ||
 (b1 && b3 && b5 && b7) ||
 (b1 && b4 && b5 && b8) ||
 (b1 && b3 && b9 && b12) ||
 (b1 && b4 && b9 && b13) ||
 (b1 && b2 && b10 && b12) ||
 (b1 && b4 && b10 && b14) ||
 (b1 && b2 && b11 && b13) ||
 (b1 && b3 && b11 && b14) ||
 (b2 && b3 && b6 && b7) ||
 (b2 && b4 && b6 && b8) ||
 (b2 && b3 && b9 && b10) ||
 (b2 && b4 && b9 && b11) ||
 (b2 && b4 && b12 && b14) ||
 (b2 && b3 && b13 && b14) ||
 (b3 && b4 && b7 && b8) ||
 (b3 && b4 && b10 && b11) ||
 (b3 && b4 && b12 && b13) ||
 (b5 && b7 && b9 && b12) ||
 (b5 && b8 && b9 && b13) ||
 (b5 && b6 && b10 && b12) ||
 (b5 && b8 && b10 && b14) ||
 (b5 && b6 && b11 && b13) ||
 (b5 && b7 && b11 && b14) ||
 (b6 && b7 && b9 && b10) ||
 (b6 && b8 && b9 && b11) ||
 (b6 && b8 && b12 && b14) ||
 (b6 && b7 && b13 && b14) ||
 (b7 && b8 && b10 && b11) ||
 (b7 && b8 && b12 && b13) ||
 (b9 && b11 && b12 && b14) ||
 (b9 && b10 && b13 && b14) ||
 (b10 && b11 && b12 && b13)) = true := by
  decide +kernel
theorem edges_explicit (G : SimpleGraph (Fin 6)) [DecidableRel G.Adj] :
 G.edgeFinset.card = (if G.Adj 0 1 then 1 else 0) + (if G.Adj 0 2 then 1 else 0) + (if G.Adj 0 3 then 1 else 0) + (if G.Adj 0 4 then 1 else 0) + (if G.Adj 0 5 then 1 else 0) + (if G.Adj 1 2 then 1 else 0) + (if G.Adj 1 3 then 1 else 0) + (if G.Adj 1 4 then 1 else 0) + (if G.Adj 1 5 then 1 else 0) + (if G.Adj 2 3 then 1 else 0) + (if G.Adj 2 4 then 1 else 0) + (if G.Adj 2 5 then 1 else 0) + (if G.Adj 3 4 then 1 else 0) + (if G.Adj 3 5 then 1 else 0) + (if G.Adj 4 5 then 1 else 0) := by
 have hh := G.sum_degrees_eq_twice_card_edges
 have hd (v : Fin 6) : G.degree v = ∑ w : Fin 6, if G.Adj v w then 1 else 0 := by
  rw [← G.card_neighborFinset_eq_degree]
  simp [SimpleGraph.neighborFinset_eq_filter, Finset.sum_boole]
 simp only [hd, Fin.sum_univ_succ, Fin.sum_univ_zero, add_zero] at hh
 change ((if G.Adj 0 0 then 1 else 0) + ((if G.Adj 0 1 then 1 else 0) + ((if G.Adj 0 2 then 1 else 0) + ((if G.Adj 0 3 then 1 else 0) + ((if G.Adj 0 4 then 1 else 0) + ((if G.Adj 0 5 then 1 else 0))))))) + (((if G.Adj 1 0 then 1 else 0) + ((if G.Adj 1 1 then 1 else 0) + ((if G.Adj 1 2 then 1 else 0) + ((if G.Adj 1 3 then 1 else 0) + ((if G.Adj 1 4 then 1 else 0) + ((if G.Adj 1 5 then 1 else 0))))))) + (((if G.Adj 2 0 then 1 else 0) + ((if G.Adj 2 1 then 1 else 0) + ((if G.Adj 2 2 then 1 else 0) + ((if G.Adj 2 3 then 1 else 0) + ((if G.Adj 2 4 then 1 else 0) + ((if G.Adj 2 5 then 1 else 0))))))) + (((if G.Adj 3 0 then 1 else 0) + ((if G.Adj 3 1 then 1 else 0) + ((if G.Adj 3 2 then 1 else 0) + ((if G.Adj 3 3 then 1 else 0) + ((if G.Adj 3 4 then 1 else 0) + ((if G.Adj 3 5 then 1 else 0))))))) + (((if G.Adj 4 0 then 1 else 0) + ((if G.Adj 4 1 then 1 else 0) + ((if G.Adj 4 2 then 1 else 0) + ((if G.Adj 4 3 then 1 else 0) + ((if G.Adj 4 4 then 1 else 0) + ((if G.Adj 4 5 then 1 else 0))))))) + (((if G.Adj 5 0 then 1 else 0) + ((if G.Adj 5 1 then 1 else 0) + ((if G.Adj 5 2 then 1 else 0) + ((if G.Adj 5 3 then 1 else 0) + ((if G.Adj 5 4 then 1 else 0) + ((if G.Adj 5 5 then 1 else 0)))))))))))) = 2 * G.edgeFinset.card at hh
 simp only [SimpleGraph.irrefl, ↓reduceIte] at hh
 simp only [G.adj_comm (1 : Fin 6) 0, G.adj_comm (2 : Fin 6) 0, G.adj_comm (3 : Fin 6) 0, G.adj_comm (4 : Fin 6) 0, G.adj_comm (5 : Fin 6) 0, G.adj_comm (2 : Fin 6) 1, G.adj_comm (3 : Fin 6) 1, G.adj_comm (4 : Fin 6) 1, G.adj_comm (5 : Fin 6) 1, G.adj_comm (3 : Fin 6) 2, G.adj_comm (4 : Fin 6) 2, G.adj_comm (5 : Fin 6) 2, G.adj_comm (4 : Fin 6) 3, G.adj_comm (5 : Fin 6) 3, G.adj_comm (5 : Fin 6) 4] at hh
 omega

def NoC4 {V : Type*} (G : SimpleGraph V) : Prop :=
 ∀ a b c d, a ≠ c → b ≠ d → G.Adj a b → G.Adj b c → G.Adj c d → ¬ G.Adj d a

theorem fin_six (G : SimpleGraph (Fin 6)) [DecidableRel G.Adj] (hfree : NoC4 G) :
 G.edgeFinset.card ≤ 7 := by
 have hb := bool_bound (decide (G.Adj 0 1)) (decide (G.Adj 0 2)) (decide (G.Adj 0 3)) (decide (G.Adj 0 4)) (decide (G.Adj 0 5)) (decide (G.Adj 1 2)) (decide (G.Adj 1 3)) (decide (G.Adj 1 4)) (decide (G.Adj 1 5)) (decide (G.Adj 2 3)) (decide (G.Adj 2 4)) (decide (G.Adj 2 5)) (decide (G.Adj 3 4)) (decide (G.Adj 3 5)) (decide (G.Adj 4 5))
 have bt (p : Prop) [Decidable p] : (decide p).toNat = if p then 1 else 0 := by
   by_cases hp : p <;> simp [hp]
 simp only [bt, Bool.or_eq_true, Bool.and_eq_true, decide_eq_true_eq, or_assoc] at hb
 rcases hb with h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h | h
 · rw [edges_explicit]
   exact h
 · rcases h with ⟨⟨⟨h0, h1⟩, h2⟩, h3⟩
   exact False.elim (hfree 0 1 2 3 (by decide) (by decide) h0 h2 h3 h1.symm)
 · rcases h with ⟨⟨⟨h0, h1⟩, h2⟩, h3⟩
   exact False.elim (hfree 0 1 2 4 (by decide) (by decide) h0 h2 h3 h1.symm)
 · rcases h with ⟨⟨⟨h0, h1⟩, h2⟩, h3⟩
   exact False.elim (hfree 0 1 2 5 (by decide) (by decide) h0 h2 h3 h1.symm)
 · rcases h with ⟨⟨⟨h0, h1⟩, h2⟩, h3⟩
-- 1487 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

/-!
Finite-order reduction for the +4 cubic-count equality case.
This is conditional on a hypothetical lexicographically minimal counterexample.
No existence or exclusion of all small graphs is asserted. Novelty unverified.
Submissions must not import this canonical module.
-/

namespace Statements.ErdosGyarfasBoundaryFourOrder

/-- `G` contains a cycle whose length is `2 ^ k` for some `k ≥ 2` (the conclusion of the root
statement `Statements.ErdosGyarfasPowerTwoCycle`). -/
def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ), c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

/-- A counterexample to the Erdős–Gyárfás conjecture: a nonempty finite simple graph with every
degree at least `3` and no power-of-two cycle. -/
def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

/-- A minimal counterexample: a counterexample that is lexicographically minimal in
(order, size) among all counterexamples on any `Fin m`. If the conjecture is true no such graph
exists, and every statement about `IsMinCex` is a statement about the structure a counterexample
would have to have. -/
def IsMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj], IsCex H →
    n < m ∨ (n = m ∧ G.edgeFinset.card ≤ H.edgeFinset.card)

/-- The canonical proposition. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj], IsMinCex G →
    3 * (Finset.univ.filter (fun v => G.degree v = 3)).card = 2 * n + 4 → n ≤ 19

/-- The target. A submission proves `statement` in its own module; the verifier bridges the two. -/
theorem target : statement := sorry

end Statements.ErdosGyarfasBoundaryFourOrder
```

### 10. A new rigidity theorem for the first boundary case left by statement 9: if a minimal counterexample has 3|V₃|…

- Permalink: https://jig.so/p/399?s=10
- Status: kernel-checked
- Filed: 2026-09-06T17:56:51.000Z by @coleski
- Version: 3

**A new rigidity theorem for the first boundary case left by statement 9: if a minimal counterexample has 3|V₃| = 2|V| + 4, then every noncubic vertex has degree exactly 4; if there are at least two noncubic vertices, at most one cubic vertex has no noncubic neighbor.**

Research update by coleski: pursuing elimination of the equality case 3c = 2n + 4, with target 3c >= 2n + 5 for a lexicographically minimal counterexample. Our proved statement 10 supplies equality structure; davidtsong independently proved the order bound n <= 19 in statement 11 and receives credit for that result. The remaining equality orders are 10, 13, 16, and 19. A local SAT encoding of the structured 10-vertex case returned UNSAT, but neither its certificate nor the graph-to-encoding bridge is yet kernel-checked. This is a work-in-progress report, not a new proved theorem or a novelty claim. The +5 target remains unproved; prior computational small-order exclusions must be credited when evaluating novelty. Formal statement and existing proof are unchanged.

**Scope.**

Hypothetical counterexamples of minimum order and then minimum size that attain equality in statement 9. The result classifies their degree structure; it does not assert that a counterexample exists or settle the Erdős–Gyárfás conjecture.

**Artifacts.**

- Cole.lean: Submissions.ErdosGyarfasBoundaryFourStructure.Cole.proof

```lean
/-
Novel continuation of Jig P399 statement 9.  That statement proves
2|V| + 4 ≤ 3|V3| for a hypothetical lexicographically minimal
Erdős–Gyárfás counterexample.  Here we classify its equality boundary:
all high-degree vertices must be quartic, and, when there are at least two of
them, at most one cubic vertex can have no high-degree neighbor.

The generic counterexample, deletion, independence, two-thirds counting, and
no-four-cycle lemmas adapt Andrew Bisch's EGC.lean (2026-08-13).  The strict
contraction argument was posted by jul059 on the Erdős problem 64 forum; the
+3 bound was proposed in Jig statement 7; the parity +4 bound is Jig statement
9.  The new ingredient is a zero-incidence refinement of statement 9's
contraction count, followed by equality and parity analysis.  This remains a
structural restriction on hypothetical counterexamples, not a resolution of
the Erdős–Gyárfás conjecture.
-/
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Walk.Maps
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Tactic

import Mathlib

namespace Submissions.ErdosGyarfasBoundaryFourStructure.Cole

namespace CycleDoubleWork

lemma interleave_nodup {A B : Type*} (l : List A) (f g : A → B)
    (hf : (l.map f).Nodup) (hg : (l.map g).Nodup)
    (hfg : ∀ x ∈ l, ∀ y ∈ l, f x ≠ g y) :
    (l.flatMap fun x => [f x, g x]).Nodup := by
  induction l with
  | nil => simp
  | cons a l ih =>
    simp only [List.map_cons, List.nodup_cons] at hf hg
    have ht := ih hf.2 hg.2 (fun x hx y hy => hfg x (by simp [hx]) y (by simp [hy]))
    simp only [List.flatMap_cons, List.cons_append, List.nil_append, List.nodup_cons]
    refine ⟨?_, ?_, ht⟩
    · simp only [List.mem_cons, List.mem_flatMap]
      intro h
      rcases h with h | ⟨x, hx, h | h⟩
      · exact hfg a (by simp) a (by simp) h
      · exact hf.1 (List.mem_map.mpr ⟨x, hx, h.symm⟩)
      · exact hfg a (by simp) x (by simp [hx]) (by simpa using h)
    · simp only [List.mem_flatMap, List.mem_cons]
      rintro ⟨x, hx, h | h⟩
      · exact hfg x (by simp [hx]) a (by simp) h.symm
      · exact hg.1 (List.mem_map.mpr ⟨x, hx, (by simpa using h : g a = g x).symm⟩)

variable {V W : Type*} {H : SimpleGraph V} {G : SimpleGraph W}

noncomputable def doubleWalk (f : V → W) (m : H.Dart → W)
    (ha : ∀ d, G.Adj (f d.fst) (m d))
    (hb : ∀ d, G.Adj (m d) (f d.snd)) :
    {u v : V} → H.Walk u v → G.Walk (f u) (f v) := by
  intro u v p
  induction p with
  | nil => exact .nil
  | @cons u v w h p ih => exact .cons (ha ⟨(u,v), h⟩) (.cons (hb ⟨(u,v), h⟩) ih)

lemma double_length (f : V → W) (m : H.Dart → W) (ha hb)
    {u v : V} (p : H.Walk u v) :
    (doubleWalk (G := G) f m ha hb p).length = 2 * p.length := by
  induction p with
  | nil => rfl
  | cons h p ih => simp only [doubleWalk] at ih; simp [doubleWalk, ih]; omega

lemma double_support (f : V → W) (m : H.Dart → W) (ha hb)
    {u v : V} (p : H.Walk u v) :
    (doubleWalk (G := G) f m ha hb p).support.tail =
      p.darts.flatMap (fun d => [m d, f d.snd]) := by
  induction p with
  | nil => rfl
  | cons h p ih =>
    simp only [doubleWalk, SimpleGraph.Walk.support_cons, List.tail_cons,
      SimpleGraph.Walk.darts_cons, List.flatMap_cons, List.cons_append, List.nil_append]
    rw [← ih, SimpleGraph.Walk.cons_tail_support]
    rfl

theorem double_cycle (f : V → W) (hf : Function.Injective f)
    (m : H.Dart → W) (ha hb) {u : V} (p : H.Walk u u)
    (hp : p.IsCycle) (hm : (p.darts.map m).Nodup)
    (hsep : ∀ d ∈ p.darts, ∀ v, m d ≠ f v) :
    (doubleWalk (G := G) f m ha hb p).IsCycle := by
  have hn : (doubleWalk f m ha hb p).support.tail.Nodup := by
    rw [double_support]
    apply interleave_nodup _ _ _ hm
    · have hh := hp.support_nodup.map hf
      rw [← SimpleGraph.Walk.map_snd_darts, List.map_map] at hh
      exact hh
    · exact fun d hd e _ => hsep d hd e.snd
  rw [SimpleGraph.Walk.isCycle_iff_isPath_tail_and_le_length]
  refine ⟨SimpleGraph.Walk.IsPath.mk' ?_, ?_⟩
  · rw [SimpleGraph.Walk.support_tail_of_not_nil]
    · exact hn
    · intro h
      have hl := (double_length f m ha hb p)
      have hp3 := hp.three_le_length
      have hz := SimpleGraph.Walk.length_eq_zero_iff.mpr h
      omega
  · rw [double_length]
    have := hp.three_le_length
    omega

theorem double_cycle_of_edge_labels (f : V → W) (hf : Function.Injective f)
    (m : H.Dart → W) (ha hb) {u : V} (p : H.Walk u u)
    (hp : p.IsCycle)
    (hm : ∀ d e, m d = m e → d.edge = e.edge)
    (hsep : ∀ d, ∀ v, m d ≠ f v) :
    (doubleWalk (G := G) f m ha hb p).IsCycle := by
  apply double_cycle f hf m ha hb p hp
  · have he := hp.isTrail.edges_nodup
    rw [SimpleGraph.Walk.edges_eq_map_darts, List.Nodup, List.pairwise_map] at he
    rw [List.Nodup, List.pairwise_map]
    exact he.imp (fun h hme => h (hm _ _ hme))
  · exact fun d _ v => hsep d v

end CycleDoubleWork
-- 1292 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

namespace Statements.ErdosGyarfasBoundaryFourStructure

def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ), c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

def IsMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj], IsCex H →
    n < m ∨ (n = m ∧ G.edgeFinset.card ≤ H.edgeFinset.card)

/-- At the first cubic-count boundary allowed by statement 9, every vertex of
degree at least four has degree exactly four. If there are at least two such
vertices, at most one cubic vertex has no neighbor of degree at least four. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj], IsMinCex G →
    3 * (Finset.univ.filter fun v => G.degree v = 3).card = 2 * n + 4 →
      (∀ u : Fin n, 4 ≤ G.degree u → G.degree u = 4) ∧
        (2 ≤ (Finset.univ.filter fun v => 4 ≤ G.degree v).card →
          ((Finset.univ.filter fun x => G.degree x = 3).filter fun x =>
            ((Finset.univ.filter fun v => 4 ≤ G.degree v).filter fun v =>
              G.Adj v x).card = 0).card ≤ 1)

theorem target : statement := by sorry

end Statements.ErdosGyarfasBoundaryFourStructure
```

### 9. In a counterexample to the Erdős–Gyárfás conjecture of minimum order and then minimum size, 3c ≥ 2n + 4, wher…

- Permalink: https://jig.so/p/399?s=9
- Status: kernel-checked
- Filed: 2026-09-06T17:26:46.000Z by @davidtsong
- Version: 2

**In a counterexample to the Erdős–Gyárfás conjecture of minimum order and then minimum size, 3c ≥ 2n + 4, where c is the number of vertices of degree three.**

Equality in the +3 bound would force odd total degree.

**Scope.**

For every n and simple graph G on Fin n satisfying IsMinCex G as defined here (nonempty, minimum degree at least three, no power-of-two cycle, lexicographically minimal in order then size), 2*n+4 ≤ 3*#{v : degree v = 3}. Hypothetical counterexamples only; no existence asserted and vacuous if the root conjecture is true.

**Artifacts.**

- CubicBoundFourWork.lean: Submissions.ErdosGyarfasMinCexCubicBoundFour.CubicBoundFourWork.proof

```lean
/-
Candidate strengthening of Jig P399 s7: a hypothetical minimal
Erdős–Gyárfás counterexample satisfies 2|V| + 4 ≤ 3|V3|.

The generic counterexample, deletion, independence, two-thirds counting, and
no-four-cycle lemmas below adapt Andrew Bisch's existing EGC.lean, commit
77c14e7c34a70530e9b88ad8160b6329a0e8e360 (2026-08-13).
The strict improvement was posted by jul059 on the Erdős problem 64 forum
on 2026-07-26. The stronger +3 argument was proposed by Jig's poser on
2026-09-04 (P399 s7). We verify that argument: the contraction graph has
empty 3-core, hence at most 2|V4|-3 edges; a refined incidence count finishes.
For |V4|≤1 we use a direct degree+2≤|V| bound instead of the n≥8 theorem.
The additional +4 step excludes equality in +3 by the handshaking parity
identity. That strengthening's novelty has not been established by this file;
prior-art review is required before any novelty claim or submission.
No counterexample is constructed, and the Erdős–Gyárfás conjecture stays open.
-/
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Walk.Maps
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Tactic

import Mathlib

namespace Submissions.ErdosGyarfasMinCexCubicBoundFour.CubicBoundFourWork

namespace CycleDoubleWork

lemma interleave_nodup {A B : Type*} (l : List A) (f g : A → B)
    (hf : (l.map f).Nodup) (hg : (l.map g).Nodup)
    (hfg : ∀ x ∈ l, ∀ y ∈ l, f x ≠ g y) :
    (l.flatMap fun x => [f x, g x]).Nodup := by
  induction l with
  | nil => simp
  | cons a l ih =>
    simp only [List.map_cons, List.nodup_cons] at hf hg
    have ht := ih hf.2 hg.2 (fun x hx y hy => hfg x (by simp [hx]) y (by simp [hy]))
    simp only [List.flatMap_cons, List.cons_append, List.nil_append, List.nodup_cons]
    refine ⟨?_, ?_, ht⟩
    · simp only [List.mem_cons, List.mem_flatMap]
      intro h
      rcases h with h | ⟨x, hx, h | h⟩
      · exact hfg a (by simp) a (by simp) h
      · exact hf.1 (List.mem_map.mpr ⟨x, hx, h.symm⟩)
      · exact hfg a (by simp) x (by simp [hx]) (by simpa using h)
    · simp only [List.mem_flatMap, List.mem_cons]
      rintro ⟨x, hx, h | h⟩
      · exact hfg x (by simp [hx]) a (by simp) h.symm
      · exact hg.1 (List.mem_map.mpr ⟨x, hx, (by simpa using h : g a = g x).symm⟩)

variable {V W : Type*} {H : SimpleGraph V} {G : SimpleGraph W}

noncomputable def doubleWalk (f : V → W) (m : H.Dart → W)
    (ha : ∀ d, G.Adj (f d.fst) (m d))
    (hb : ∀ d, G.Adj (m d) (f d.snd)) :
    {u v : V} → H.Walk u v → G.Walk (f u) (f v) := by
  intro u v p
  induction p with
  | nil => exact .nil
  | @cons u v w h p ih => exact .cons (ha ⟨(u,v), h⟩) (.cons (hb ⟨(u,v), h⟩) ih)

lemma double_length (f : V → W) (m : H.Dart → W) (ha hb)
    {u v : V} (p : H.Walk u v) :
    (doubleWalk (G := G) f m ha hb p).length = 2 * p.length := by
  induction p with
  | nil => rfl
  | cons h p ih => simp only [doubleWalk] at ih; simp [doubleWalk, ih]; omega

lemma double_support (f : V → W) (m : H.Dart → W) (ha hb)
    {u v : V} (p : H.Walk u v) :
    (doubleWalk (G := G) f m ha hb p).support.tail =
      p.darts.flatMap (fun d => [m d, f d.snd]) := by
  induction p with
  | nil => rfl
  | cons h p ih =>
    simp only [doubleWalk, SimpleGraph.Walk.support_cons, List.tail_cons,
      SimpleGraph.Walk.darts_cons, List.flatMap_cons, List.cons_append, List.nil_append]
    rw [← ih, SimpleGraph.Walk.cons_tail_support]
    rfl

theorem double_cycle (f : V → W) (hf : Function.Injective f)
    (m : H.Dart → W) (ha hb) {u : V} (p : H.Walk u u)
    (hp : p.IsCycle) (hm : (p.darts.map m).Nodup)
    (hsep : ∀ d ∈ p.darts, ∀ v, m d ≠ f v) :
    (doubleWalk (G := G) f m ha hb p).IsCycle := by
  have hn : (doubleWalk f m ha hb p).support.tail.Nodup := by
    rw [double_support]
    apply interleave_nodup _ _ _ hm
    · have hh := hp.support_nodup.map hf
      rw [← SimpleGraph.Walk.map_snd_darts, List.map_map] at hh
      exact hh
    · exact fun d hd e _ => hsep d hd e.snd
  rw [SimpleGraph.Walk.isCycle_iff_isPath_tail_and_le_length]
  refine ⟨SimpleGraph.Walk.IsPath.mk' ?_, ?_⟩
  · rw [SimpleGraph.Walk.support_tail_of_not_nil]
    · exact hn
    · intro h
      have hl := (double_length f m ha hb p)
      have hp3 := hp.three_le_length
      have hz := SimpleGraph.Walk.length_eq_zero_iff.mpr h
      omega
  · rw [double_length]
    have := hp.three_le_length
    omega

theorem double_cycle_of_edge_labels (f : V → W) (hf : Function.Injective f)
    (m : H.Dart → W) (ha hb) {u : V} (p : H.Walk u u)
    (hp : p.IsCycle)
    (hm : ∀ d e, m d = m e → d.edge = e.edge)
    (hsep : ∀ d, ∀ v, m d ≠ f v) :
    (doubleWalk (G := G) f m ha hb p).IsCycle := by
  apply double_cycle f hf m ha hb p hp
  · have he := hp.isTrail.edges_nodup
    rw [SimpleGraph.Walk.edges_eq_map_darts, List.Nodup, List.pairwise_map] at he
    rw [List.Nodup, List.pairwise_map]
    exact he.imp (fun h hme => h (hm _ _ hme))
  · exact fun d _ v => hsep d v

-- 1160 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

/-!
A parity strengthening of P399 s7 for hypothetical minimal counterexamples.
The +3 equality forces odd total degree, contradicting handshaking.
This does not construct a counterexample or settle the root. Novelty unverified.
Submissions must not import this module; target uses sorry.
-/

namespace Statements.ErdosGyarfasMinCexCubicBoundFour

/-- `G` contains a cycle whose length is `2 ^ k` for some `k ≥ 2` (the conclusion of the root
statement `Statements.ErdosGyarfasPowerTwoCycle`). -/
def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ), c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

/-- A counterexample to the Erdős–Gyárfás conjecture: a nonempty finite simple graph with every
degree at least `3` and no power-of-two cycle. -/
def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

/-- A minimal counterexample: a counterexample that is lexicographically minimal in
(order, size) among all counterexamples on any `Fin m`. If the conjecture is true no such graph
exists, and every statement about `IsMinCex` is a statement about the structure a counterexample
would have to have. -/
def IsMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj], IsCex H →
    n < m ∨ (n = m ∧ G.edgeFinset.card ≤ H.edgeFinset.card)

/-- The canonical proposition. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj], IsMinCex G →
    2 * n + 4 ≤ 3 * (Finset.univ.filter (fun v => G.degree v = 3)).card

/-- The target. A submission proves `statement` in its own module; the verifier bridges the two. -/
theorem target : statement := sorry

end Statements.ErdosGyarfasMinCexCubicBoundFour
```

### 8. For every fixed end margin, a serial arithmetic spectrum with thirteen consecutive residue classes modulo 17…

- Permalink: https://jig.so/p/399?s=8
- Status: kernel-checked
- Filed: 2026-09-06T00:33:10.000Z by @declangessel
- Version: 2

**For every fixed end margin, a serial arithmetic spectrum with thirteen consecutive residue classes modulo 17 can span a power of two yet contain no power of two, even though its residue classes meet the doubling orbit.**

The increment count and both margins can be arbitrarily large.

**Scope.**

For every natural end-margin bound B, spectra S_d={17(12d+t)+r: t≤8d, r≤12} with d≥1 and doubling modulus 17.

**Artifacts.**

- Declan.lean: Submissions.ErdosGyarfasOrbitSpanningBarrier.Declan.proof

```lean
import Mathlib.Data.Nat.ModEq
import Mathlib.Tactic

/-
An obstruction to the numerical inference in Guillem Duran-Ballester,
'A Structural Exhaustion Proof of the Erdos-Gyarfas Conjecture on Power-of-Two
Cycles', Zenodo 22019344 (2026-08-20), Lemma 'System increment arithmetic'.
This is an arithmetic countermodel, not an Erdos-Gyarfas graph counterexample.
-/
namespace Submissions.ErdosGyarfasOrbitSpanningBarrier.Declan

/-- The spectrum of a serial system with increment 17 and thirteen offsets. -/
def InSpectrum (d y : ℕ) : Prop :=
  ∃ t ≤ 8 * d, ∃ r ≤ 12, y = 17 * (12 * d + t) + r

/-- A long central spectral interval can miss every power of two even though
its residue classes intersect the complete doubling orbit modulo 17. -/
theorem no_power_in_spectrum {d k : ℕ} (hd : 1 ≤ d)
    (hk : 2 ^ k = 272 * d + 16) :
    ∀ j : ℕ, ¬ InSpectrum d (2 ^ j) := by
  intro j ⟨t, ht, r, hr, he⟩
  have hlo : 2 ^ k < 2 * 2 ^ j := by omega
  have hhi : 2 ^ j < 2 * 2 ^ k := by omega
  have heq : j = k := by
    rcases lt_trichotomy j k with hlt | heq | hgt
    · have hp : 2 ^ (j + 1) ≤ 2 ^ k :=
        pow_le_pow_right' (by omega : 1 ≤ (2 : ℕ)) (by omega)
      rw [pow_succ] at hp
      omega
    · exact heq
    · have hp : 2 ^ (k + 1) ≤ 2 ^ j :=
        pow_le_pow_right' (by omega : 1 ≤ (2 : ℕ)) (by omega)
      rw [pow_succ] at hp
      omega
  subst j
  omega

theorem spans_power {d k : ℕ} (hd : 1 ≤ d)
    (hk : 2 ^ k = 272 * d + 16) :
    204 * d < 2 ^ k ∧ 2 ^ k < 340 * d + 12 := by omega

/-- These are the two realized endpoint lengths of the spectrum. -/
theorem spectral_endpoints (d : ℕ) :
    InSpectrum d (204 * d) ∧ InSpectrum d (340 * d + 12) := by
  constructor
  · exact ⟨0, by omega, 0, by omega, by omega⟩
  · exact ⟨8 * d, by omega, 12, by omega, by omega⟩

/-- The doubling orbit has order eight modulo 17. -/
theorem orbit_certificate :
    2 ^ 8 % 17 = 1 ∧
    (∀ k ∈ Finset.Icc 1 7, 2 ^ k % 17 ≠ 1) ∧
    8 > 17 - 13 ∧
    (∃ k : ℕ, 2 ^ k % 17 ≤ 12) := by
  refine ⟨by norm_num, by decide, by norm_num, 0, by norm_num⟩

/-- Explicit finite countermodel to orbit-hit + scale-spanning => power-hit. -/
theorem concrete_countermodel :
    204 * 3855 < 2 ^ 20 ∧
    2 ^ 20 < 340 * 3855 + 12 ∧
    InSpectrum 3855 (204 * 3855) ∧
    InSpectrum 3855 (340 * 3855 + 12) ∧
    (∀ j : ℕ, ¬ InSpectrum 3855 (2 ^ j)) ∧
    8 * 3855 ≥ 2 * 17 ^ 2 := by
  have hk : (2 : ℕ) ^ 20 = 272 * 3855 + 16 := by norm_num
  have hs := spans_power (d := 3855) (by omega) hk
  have he := spectral_endpoints 3855
  exact ⟨hs.1, hs.2, he.1, he.2,
    no_power_in_spectrum (by omega) hk, by norm_num⟩

/-- Arbitrarily large members exist; bounded end trimming cannot repair the
inference. Every eighth binary exponent supplies the same forbidden residue. -/
theorem arbitrarily_large_parameters (B : ℕ) :
    ∃ d k : ℕ, B ≤ d ∧ 1 ≤ d ∧ 2 ^ k = 272 * d + 16 := by
  obtain ⟨n, hn⟩ := pow_unbounded_of_one_lt (272 * (B + 1) + 16)
    (by norm_num : 1 < (256 : ℕ))
  have hz : 256 ^ n % 17 = 1 := by norm_num [Nat.pow_mod]
  let d := (256 ^ n - 1) / 17
  have hd : 256 ^ n = 17 * d + 1 := by
    have hm := Nat.mod_add_div (256 ^ n - 1) 17
    have hp : 1 ≤ 256 ^ n := Nat.one_le_pow _ _ (by omega)
    dsimp [d]
    omega
  refine ⟨d, 8 * n + 4, by omega, by omega, ?_⟩
  calc
    (2 : ℕ) ^ (8 * n + 4) = 256 ^ n * 16 := by
      rw [pow_add, pow_mul]
      norm_num
    _ = 272 * d + 16 := by omega

/-- The power lies arbitrarily far from both spectrum endpoints, and there
are arbitrarily many identical increments. A fixed end correction or fixed
frequency threshold therefore cannot validate the orbit-hit inference. -/
theorem robust_countermodels (B : ℕ) :
    ∃ d k : ℕ,
      1 ≤ d ∧ B ≤ 8 * d ∧
      204 * d + 17 * B + 12 < 2 ^ k ∧
      2 ^ k + 17 * B + 12 < 340 * d + 12 ∧
      InSpectrum d (204 * d) ∧
      InSpectrum d (340 * d + 12) ∧
      (∀ j : ℕ, ¬ InSpectrum d (2 ^ j)) := by
  obtain ⟨d, k, hdB, hd, hk⟩ := arbitrarily_large_parameters (B + 1)
  have he := spectral_endpoints d
  exact ⟨d, k, hd, by omega, by omega, by omega, he.1, he.2,
    no_power_in_spectrum hd hk⟩

theorem no_power_between {k y : ℕ} (hlo : 2 ^ k < y)
    (hhi : y < 2 * 2 ^ k) : ∀ j : ℕ, 2 ^ j ≠ y := by
  intro j he
  by_cases hj : j ≤ k
  · have hp := pow_le_pow_right' (by omega : 1 ≤ (2 : ℕ)) hj
    omega
  · have hp : 2 ^ (k + 1) ≤ 2 ^ j :=
      pow_le_pow_right' (by omega : 1 ≤ (2 : ℕ)) (by omega)
    rw [pow_succ] at hp
    omega

/-- The other two kinds of cycle lengths in the explicit serial graph
realization are also harmless: a single cell has length21; two closing paths
have combined length376d+r+s. The graph realization itself is explained in
-- 30 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Nat.ModEq
import Mathlib.Tactic

/-
A numerical barrier to the modular orbit / dyadic-spanning inference proposed
in Duran-Ballester, Zenodo22019344, 'System increment arithmetic', part(a).
This statement concerns finite arithmetic spectra only. It neither gives a
graph counterexample to Erdos64 nor refutes all extra residual hypotheses of
that paper's full lemma. Its arbitrary parameterB excludes repair by any
fixed frequency threshold or fixed end trimming.
-/
namespace Statements.ErdosGyarfasOrbitSpanningBarrier

def InSpectrum (d y : ℕ) : Prop :=
  ∃ t ≤ 8 * d, ∃ r ≤ 12, y = 17 * (12 * d + t) + r

abbrev statement : Prop :=
  (2 ^ 8 % 17 = 1 ∧
    (∀ k ∈ Finset.Icc 1 7, 2 ^ k % 17 ≠ 1) ∧
    8 > 17 - 13 ∧ (∃ k : ℕ, 2 ^ k % 17 ≤ 12)) ∧
  ∀ B : ℕ, ∃ d k : ℕ,
    1 ≤ d ∧ B ≤ 8 * d ∧
    204 * d + 17 * B + 12 < 2 ^ k ∧
    2 ^ k + 17 * B + 12 < 340 * d + 12 ∧
    InSpectrum d (204 * d) ∧
    InSpectrum d (340 * d + 12) ∧
    (∀ j : ℕ, ¬ InSpectrum d (2 ^ j))

-- The open canonical target; submissions must not import this module.
theorem target : statement := sorry

end Statements.ErdosGyarfasOrbitSpanningBarrier
```

### 7. In a counterexample to the Erdős–Gyárfás conjecture of minimum order and then minimum size, 3|V3| ≥ 2n + 3, b…

- Permalink: https://jig.so/p/399?s=7
- Status: jig-cited
- Filed: 2026-09-04T23:02:58.000Z by @am00lya, @davidtsong / Fable 5.1 / Claude Code
- Version: 2

**In a counterexample to the Erdős–Gyárfás conjecture of minimum order and then minimum size, 3|V3| ≥ 2n + 3, because the contraction graph on the high-degree vertices must be 2-degenerate.**

**Scope.**

Hypothetical minimal counterexamples only: graphs G on Fin n with IsMinCex G as defined in the statement; conclusion 2n + 3 ≤ 3|{v : deg v = 3}|, sharpening the strict bound 2n + 1 ≤ 3|V3| (?s=6) by the empty-3-core property of the contraction graph H on V4; vacuous if the conjecture is true.

**Artifacts.**

- CubicBoundThreeWork.lean: Submissions.ErdosGyarfasMinCexCubicBoundThree.CubicBoundThreeWork.proof

```lean
/-
Jig P399 s7: a hypothetical minimal Erdős–Gyárfás counterexample satisfies
2|V| + 3 ≤ 3|V3|, where V3 denotes its cubic vertices.

The generic counterexample, deletion, independence, two-thirds counting, and
no-four-cycle lemmas below adapt Andrew Bisch's existing EGC.lean, commit
77c14e7c34a70530e9b88ad8160b6329a0e8e360 (2026-08-13).
The strict improvement was posted by jul059 on the Erdős problem 64 forum
on 2026-07-26. The stronger +3 argument was proposed by Jig's poser on
2026-09-04 (P399 s7). We verify that argument: the contraction graph has
empty 3-core, hence at most 2|V4|-3 edges; a refined incidence count finishes.
For |V4|≤1 we use a direct degree+2≤|V| bound instead of the n≥8 theorem.
This verifies existing informal arguments, not a new mathematical discovery.
No counterexample is constructed, and the Erdős–Gyárfás conjecture stays open.
-/
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Walk.Maps
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Tactic

import Mathlib

namespace Submissions.ErdosGyarfasMinCexCubicBoundThree.CubicBoundThreeWork

namespace CycleDoubleWork

lemma interleave_nodup {A B : Type*} (l : List A) (f g : A → B)
    (hf : (l.map f).Nodup) (hg : (l.map g).Nodup)
    (hfg : ∀ x ∈ l, ∀ y ∈ l, f x ≠ g y) :
    (l.flatMap fun x => [f x, g x]).Nodup := by
  induction l with
  | nil => simp
  | cons a l ih =>
    simp only [List.map_cons, List.nodup_cons] at hf hg
    have ht := ih hf.2 hg.2 (fun x hx y hy => hfg x (by simp [hx]) y (by simp [hy]))
    simp only [List.flatMap_cons, List.cons_append, List.nil_append, List.nodup_cons]
    refine ⟨?_, ?_, ht⟩
    · simp only [List.mem_cons, List.mem_flatMap]
      intro h
      rcases h with h | ⟨x, hx, h | h⟩
      · exact hfg a (by simp) a (by simp) h
      · exact hf.1 (List.mem_map.mpr ⟨x, hx, h.symm⟩)
      · exact hfg a (by simp) x (by simp [hx]) (by simpa using h)
    · simp only [List.mem_flatMap, List.mem_cons]
      rintro ⟨x, hx, h | h⟩
      · exact hfg x (by simp [hx]) a (by simp) h.symm
      · exact hg.1 (List.mem_map.mpr ⟨x, hx, (by simpa using h : g a = g x).symm⟩)

variable {V W : Type*} {H : SimpleGraph V} {G : SimpleGraph W}

noncomputable def doubleWalk (f : V → W) (m : H.Dart → W)
    (ha : ∀ d, G.Adj (f d.fst) (m d))
    (hb : ∀ d, G.Adj (m d) (f d.snd)) :
    {u v : V} → H.Walk u v → G.Walk (f u) (f v) := by
  intro u v p
  induction p with
  | nil => exact .nil
  | @cons u v w h p ih => exact .cons (ha ⟨(u,v), h⟩) (.cons (hb ⟨(u,v), h⟩) ih)

lemma double_length (f : V → W) (m : H.Dart → W) (ha hb)
    {u v : V} (p : H.Walk u v) :
    (doubleWalk (G := G) f m ha hb p).length = 2 * p.length := by
  induction p with
  | nil => rfl
  | cons h p ih => simp only [doubleWalk] at ih; simp [doubleWalk, ih]; omega

lemma double_support (f : V → W) (m : H.Dart → W) (ha hb)
    {u v : V} (p : H.Walk u v) :
    (doubleWalk (G := G) f m ha hb p).support.tail =
      p.darts.flatMap (fun d => [m d, f d.snd]) := by
  induction p with
  | nil => rfl
  | cons h p ih =>
    simp only [doubleWalk, SimpleGraph.Walk.support_cons, List.tail_cons,
      SimpleGraph.Walk.darts_cons, List.flatMap_cons, List.cons_append, List.nil_append]
    rw [← ih, SimpleGraph.Walk.cons_tail_support]
    rfl

theorem double_cycle (f : V → W) (hf : Function.Injective f)
    (m : H.Dart → W) (ha hb) {u : V} (p : H.Walk u u)
    (hp : p.IsCycle) (hm : (p.darts.map m).Nodup)
    (hsep : ∀ d ∈ p.darts, ∀ v, m d ≠ f v) :
    (doubleWalk (G := G) f m ha hb p).IsCycle := by
  have hn : (doubleWalk f m ha hb p).support.tail.Nodup := by
    rw [double_support]
    apply interleave_nodup _ _ _ hm
    · have hh := hp.support_nodup.map hf
      rw [← SimpleGraph.Walk.map_snd_darts, List.map_map] at hh
      exact hh
    · exact fun d hd e _ => hsep d hd e.snd
  rw [SimpleGraph.Walk.isCycle_iff_isPath_tail_and_le_length]
  refine ⟨SimpleGraph.Walk.IsPath.mk' ?_, ?_⟩
  · rw [SimpleGraph.Walk.support_tail_of_not_nil]
    · exact hn
    · intro h
      have hl := (double_length f m ha hb p)
      have hp3 := hp.three_le_length
      have hz := SimpleGraph.Walk.length_eq_zero_iff.mpr h
      omega
  · rw [double_length]
    have := hp.three_le_length
    omega

theorem double_cycle_of_edge_labels (f : V → W) (hf : Function.Injective f)
    (m : H.Dart → W) (ha hb) {u : V} (p : H.Walk u u)
    (hp : p.IsCycle)
    (hm : ∀ d e, m d = m e → d.edge = e.edge)
    (hsep : ∀ d, ∀ v, m d ≠ f v) :
    (doubleWalk (G := G) f m ha hb p).IsCycle := by
  apply double_cycle f hf m ha hb p hp
  · have he := hp.isTrail.edges_nodup
    rw [SimpleGraph.Walk.edges_eq_map_darts, List.Nodup, List.pairwise_map] at he
    rw [List.Nodup, List.pairwise_map]
    exact he.imp (fun h hme => h (hm _ _ hme))
  · exact fun d _ v => hsep d v

end CycleDoubleWork

-- 1042 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

/-!
# ErdosGyarfasMinCexCubicBoundThree — the contraction graph of a minimal counterexample is 2-degenerate

Structure of a hypothetical minimal counterexample to the Erdős–Gyárfás conjecture (Jig problem
399, Erdős #64): if `G` is a counterexample of minimum order and then minimum size, then
`2 n + 3 ≤ 3 |{v : deg v = 3}|`. This sharpens the strict two-thirds bound
(`Statements.ErdosGyarfasMinCexCubicStrict`, `2 n + 1 ≤ 3 |V₃|`) by the following mechanism.
Let `V₄` be the (independent) set of vertices of degree at least `4` and `H` the simple graph on
`V₄` with one edge `u u'` for every cubic vertex `x` whose two high-degree neighbours are `u, u'`
(distinct cubic vertices give distinct pairs, since `G` has no 4-cycle). Every cycle of `H`
lifts to a cycle of twice the length in `G`, so no nonempty induced subgraph of `H` can have all
degrees at least `3` (it would be a smaller graph with `δ ≥ 3`, hence by minimality it would
contain a `2^k`-cycle, which lifts to a `2^(k+1)`-cycle of `G`). Thus `H` is 2-degenerate and
has at most `2 |V₄| - 3` edges when `|V₄| ≥ 2`. Counting the edges between `V₄` and the cubic
vertices then gives `|V₃| ≥ 2 |V₄| + 3`; the cases `|V₄| ≤ 1` follow from `n ≥ 8`
(`Statements.ErdosGyarfasSmallOrderC4`). Proposed by the poser on 2026-09-04 with this informal
proof; not yet kernel-checked. Vacuous if the conjecture is true.

Submissions **must not** import this module: `target` below is closed with `sorry`.
-/

namespace Statements.ErdosGyarfasMinCexCubicBoundThree

/-- `G` contains a cycle whose length is `2 ^ k` for some `k ≥ 2` (the conclusion of the root
statement `Statements.ErdosGyarfasPowerTwoCycle`). -/
def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ), c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

/-- A counterexample to the Erdős–Gyárfás conjecture: a nonempty finite simple graph with every
degree at least `3` and no power-of-two cycle. -/
def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

/-- A minimal counterexample: a counterexample that is lexicographically minimal in
(order, size) among all counterexamples on any `Fin m`. If the conjecture is true no such graph
exists, and every statement about `IsMinCex` is a statement about the structure a counterexample
would have to have. -/
def IsMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj], IsCex H →
    n < m ∨ (n = m ∧ G.edgeFinset.card ≤ H.edgeFinset.card)

/-- The canonical proposition. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj], IsMinCex G →
    2 * n + 3 ≤ 3 * (Finset.univ.filter (fun v => G.degree v = 3)).card

/-- The target. A submission proves `statement` in its own module; the verifier bridges the two. -/
theorem target : statement := sorry

end Statements.ErdosGyarfasMinCexCubicBoundThree
```

### 6. In a counterexample to the Erdős–Gyárfás conjecture of minimum order and then minimum size, strictly more tha…

- Permalink: https://jig.so/p/399?s=6
- Status: prior art
- Filed: 2026-09-04T23:02:50.000Z by @am00lya, @davidtsong / Fable 5.1 / Claude Code
- Version: 2

**In a counterexample to the Erdős–Gyárfás conjecture of minimum order and then minimum size, strictly more than two thirds of the vertices have degree exactly 3.**

**Scope.**

Hypothetical minimal counterexamples only: graphs G on Fin n with IsMinCex G as defined in the statement (0 < n, all degrees ≥ 3, no cycle of length 2^k with k ≥ 2, lexicographically minimal in (order, number of edges) over all Fin m); conclusion 2n < 3|{v : deg v = 3}|, which strengthens Bisch's 2n ≤ 3|V3| and Carr's 4/7; vacuous if the conjecture is true.

**Artifacts.**

- CubicStrictWork.lean: Submissions.ErdosGyarfasMinCexCubicStrict.CubicStrictWork.proof

```lean
/-
Jig P399 s6: strictly more than two thirds of the vertices in a hypothetical
minimal Erdős–Gyárfás counterexample are cubic.

The generic counterexample, deletion, independence, two-thirds counting, and
no-four-cycle lemmas below adapt Andrew Bisch's existing EGC.lean, commit
77c14e7c34a70530e9b88ad8160b6329a0e8e360 (2026-08-13).
The strict improvement is the argument posted by jul059 on the Erdős problem
64 forum on 2026-07-26, credited there to ChatGPT 5.6 Sol High and unverified.
We formalize its equality case, smaller contraction graph, and cycle doubling.
This verifies an existing informal argument, not a new mathematical discovery.
No counterexample is constructed, and the Erdős–Gyárfás conjecture stays open.
-/
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Walk.Maps
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Tactic

import Mathlib

namespace Submissions.ErdosGyarfasMinCexCubicStrict.CubicStrictWork

namespace CycleDoubleWork

lemma interleave_nodup {A B : Type*} (l : List A) (f g : A → B)
    (hf : (l.map f).Nodup) (hg : (l.map g).Nodup)
    (hfg : ∀ x ∈ l, ∀ y ∈ l, f x ≠ g y) :
    (l.flatMap fun x => [f x, g x]).Nodup := by
  induction l with
  | nil => simp
  | cons a l ih =>
    simp only [List.map_cons, List.nodup_cons] at hf hg
    have ht := ih hf.2 hg.2 (fun x hx y hy => hfg x (by simp [hx]) y (by simp [hy]))
    simp only [List.flatMap_cons, List.cons_append, List.nil_append, List.nodup_cons]
    refine ⟨?_, ?_, ht⟩
    · simp only [List.mem_cons, List.mem_flatMap]
      intro h
      rcases h with h | ⟨x, hx, h | h⟩
      · exact hfg a (by simp) a (by simp) h
      · exact hf.1 (List.mem_map.mpr ⟨x, hx, h.symm⟩)
      · exact hfg a (by simp) x (by simp [hx]) (by simpa using h)
    · simp only [List.mem_flatMap, List.mem_cons]
      rintro ⟨x, hx, h | h⟩
      · exact hfg x (by simp [hx]) a (by simp) h.symm
      · exact hg.1 (List.mem_map.mpr ⟨x, hx, (by simpa using h : g a = g x).symm⟩)

variable {V W : Type*} {H : SimpleGraph V} {G : SimpleGraph W}

noncomputable def doubleWalk (f : V → W) (m : H.Dart → W)
    (ha : ∀ d, G.Adj (f d.fst) (m d))
    (hb : ∀ d, G.Adj (m d) (f d.snd)) :
    {u v : V} → H.Walk u v → G.Walk (f u) (f v) := by
  intro u v p
  induction p with
  | nil => exact .nil
  | @cons u v w h p ih => exact .cons (ha ⟨(u,v), h⟩) (.cons (hb ⟨(u,v), h⟩) ih)

lemma double_length (f : V → W) (m : H.Dart → W) (ha hb)
    {u v : V} (p : H.Walk u v) :
    (doubleWalk (G := G) f m ha hb p).length = 2 * p.length := by
  induction p with
  | nil => rfl
  | cons h p ih => simp only [doubleWalk] at ih; simp [doubleWalk, ih]; omega

lemma double_support (f : V → W) (m : H.Dart → W) (ha hb)
    {u v : V} (p : H.Walk u v) :
    (doubleWalk (G := G) f m ha hb p).support.tail =
      p.darts.flatMap (fun d => [m d, f d.snd]) := by
  induction p with
  | nil => rfl
  | cons h p ih =>
    simp only [doubleWalk, SimpleGraph.Walk.support_cons, List.tail_cons,
      SimpleGraph.Walk.darts_cons, List.flatMap_cons, List.cons_append, List.nil_append]
    rw [← ih, SimpleGraph.Walk.cons_tail_support]
    rfl

theorem double_cycle (f : V → W) (hf : Function.Injective f)
    (m : H.Dart → W) (ha hb) {u : V} (p : H.Walk u u)
    (hp : p.IsCycle) (hm : (p.darts.map m).Nodup)
    (hsep : ∀ d ∈ p.darts, ∀ v, m d ≠ f v) :
    (doubleWalk (G := G) f m ha hb p).IsCycle := by
  have hn : (doubleWalk f m ha hb p).support.tail.Nodup := by
    rw [double_support]
    apply interleave_nodup _ _ _ hm
    · have hh := hp.support_nodup.map hf
      rw [← SimpleGraph.Walk.map_snd_darts, List.map_map] at hh
      exact hh
    · exact fun d hd e _ => hsep d hd e.snd
  rw [SimpleGraph.Walk.isCycle_iff_isPath_tail_and_le_length]
  refine ⟨SimpleGraph.Walk.IsPath.mk' ?_, ?_⟩
  · rw [SimpleGraph.Walk.support_tail_of_not_nil]
    · exact hn
    · intro h
      have hl := (double_length f m ha hb p)
      have hp3 := hp.three_le_length
      have hz := SimpleGraph.Walk.length_eq_zero_iff.mpr h
      omega
  · rw [double_length]
    have := hp.three_le_length
    omega

theorem double_cycle_of_edge_labels (f : V → W) (hf : Function.Injective f)
    (m : H.Dart → W) (ha hb) {u : V} (p : H.Walk u u)
    (hp : p.IsCycle)
    (hm : ∀ d e, m d = m e → d.edge = e.edge)
    (hsep : ∀ d, ∀ v, m d ≠ f v) :
    (doubleWalk (G := G) f m ha hb p).IsCycle := by
  apply double_cycle f hf m ha hb p hp
  · have he := hp.isTrail.edges_nodup
    rw [SimpleGraph.Walk.edges_eq_map_darts, List.Nodup, List.pairwise_map] at he
    rw [List.Nodup, List.pairwise_map]
    exact he.imp (fun h hme => h (hm _ _ hme))
  · exact fun d _ v => hsep d v

end CycleDoubleWork

open Finset SimpleGraph

-- 723 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

/-!
# ErdosGyarfasMinCexCubicStrict — strictly more than two thirds of a minimal counterexample is cubic

Structure of a hypothetical minimal counterexample to the Erdős–Gyárfás conjecture (Jig problem
399, Erdős #64): if `G` is a counterexample of minimum order and then minimum size, then
`2 n < 3 |{v : deg v = 3}|`. Counting edges between the cubic vertices and the vertices of degree
at least `4` (which are independent, and every vertex has a cubic neighbour) gives
`2 n ≤ 3 |V₃|` (Carr: `4/7`; Bisch, Zenodo 10.5281/zenodo.21574476: `2/3`). Equality would force
every high-degree vertex to have degree exactly `4` and every cubic vertex to have exactly two
high-degree neighbours; contracting each cubic vertex to an edge between its two high-degree
neighbours then yields a simple 4-regular graph on fewer vertices, whose power-of-two cycle
(forced by minimality) lifts to a cycle of twice the length in `G`. The strict inequality was
proposed in a forum comment on erdosproblems.com/64 (26 July 2026, unverified there). Vacuous if
the conjecture is true.

Submissions **must not** import this module: `target` below is closed with `sorry`.
-/

namespace Statements.ErdosGyarfasMinCexCubicStrict

/-- `G` contains a cycle whose length is `2 ^ k` for some `k ≥ 2` (the conclusion of the root
statement `Statements.ErdosGyarfasPowerTwoCycle`). -/
def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ), c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

/-- A counterexample to the Erdős–Gyárfás conjecture: a nonempty finite simple graph with every
degree at least `3` and no power-of-two cycle. -/
def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

/-- A minimal counterexample: a counterexample that is lexicographically minimal in
(order, size) among all counterexamples on any `Fin m`. If the conjecture is true no such graph
exists, and every statement about `IsMinCex` is a statement about the structure a counterexample
would have to have. -/
def IsMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj], IsCex H →
    n < m ∨ (n = m ∧ G.edgeFinset.card ≤ H.edgeFinset.card)

/-- The canonical proposition. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj], IsMinCex G →
    2 * n < 3 * (Finset.univ.filter (fun v => G.degree v = 3)).card

/-- The target. A submission proves `statement` in its own module; the verifier bridges the two. -/
theorem target : statement := sorry

end Statements.ErdosGyarfasMinCexCubicStrict
```

### 5. In a counterexample to the Erdős–Gyárfás conjecture of minimum order and then minimum size, every vertex has…

- Permalink: https://jig.so/p/399?s=5
- Status: prior art
- Filed: 2026-09-04T22:36:34.000Z by @am00lya, @davidtsong / Fable 5.1 / Claude Code
- Version: 2

**In a counterexample to the Erdős–Gyárfás conjecture of minimum order and then minimum size, every vertex has a neighbour of degree exactly 3.**

**Scope.**

Hypothetical minimal counterexamples only: graphs G on Fin n with IsMinCex G as defined in the statement (0 < n, all degrees ≥ 3, no cycle of length 2^k with k ≥ 2, lexicographically minimal in (order, number of edges) among all such graphs on any Fin m); conclusion: every vertex is adjacent to a cubic vertex; vacuous if the conjecture is true.

**Artifacts.**

- VertexDeletion.lean: Submissions.ErdosGyarfasMinCexCubicNeighbour.VertexDeletion.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Tactic

/-!
Jig problem 399, statement 5: the known degree-three-neighbour property
of a hypothetical minimum-order counterexample to the Erdős–Gyárfás conjecture.
The proof deletes a vertex and uses only minimum order, not edge minimality.
It neither constructs a counterexample nor settles the parent conjecture.
The result is in Carr, arXiv:2605.22844, and Andrew Bisch's existing Lean
formalization EGC.IsMinCex.exists_cubic_neighbor. This file adapts the known
argument to Jig's Fin-indexed canonical proposition.
-/

namespace Submissions.ErdosGyarfasMinCexCubicNeighbour.VertexDeletion

def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ), c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

def IsMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj], IsCex H →
    n < m ∨ (n = m ∧ G.edgeFinset.card ≤ H.edgeFinset.card)

theorem cycle_of_injective_hom {n m : ℕ} {G : SimpleGraph (Fin n)}
    {H : SimpleGraph (Fin m)} (f : H →g G) (hf : Function.Injective f) :
    HasPow2Cycle H → HasPow2Cycle G := by
  rintro ⟨v, c, k, hc, hk, hlen⟩
  exact ⟨f v, c.map f, k, hc.map hf, hk, by simpa using hlen⟩

theorem survivors_card_lt {n : ℕ} (u : Fin n) :
    Fintype.card {v : Fin n // v ≠ u} < n := by
  simpa using (Fintype.card_subtype_lt (p := fun v : Fin n => v ≠ u)
    (x := u) (by simp))

theorem survivors_card_pos {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj]
    (u : Fin n) (hdeg : 3 ≤ G.degree u) :
    0 < Fintype.card {v : Fin n // v ≠ u} := by
  obtain ⟨v, hv⟩ := (G.degree_pos_iff_exists_adj u).mp (by omega : 0 < G.degree u)
  exact Fintype.card_pos_iff.mpr ⟨⟨v, hv.ne.symm⟩⟩

section Deletion

variable {V : Type*} [Fintype V] [DecidableEq V]
  (G : SimpleGraph V) [DecidableRel G.Adj] (u : V)

theorem degree_delete_eq_erase (v : {x : V // x ≠ u}) :
    (G.induce {x : V | x ≠ u}).degree v = (G.neighborFinset v.val |>.erase u).card := by
  have h := congrArg Finset.card (G.map_neighborFinset_induce (s := {x : V | x ≠ u}) v)
  simp only [Finset.card_map, SimpleGraph.card_neighborFinset_eq_degree] at h
  rw [h]
  congr 1
  ext x
  simp [and_comm]

theorem degree_delete_ge_three
    (hdeg : ∀ v : V, 3 ≤ G.degree v)
    (hneigh : ∀ v : V, G.Adj u v → 4 ≤ G.degree v)
    (v : {x : V // x ≠ u}) :
    3 ≤ (G.induce {x : V | x ≠ u}).degree v := by
  rw [degree_delete_eq_erase]
  by_cases h : G.Adj u v.val
  · have hu : u ∈ G.neighborFinset v.val := by simpa using h.symm
    rw [Finset.card_erase_of_mem hu, SimpleGraph.card_neighborFinset_eq_degree]
    have := hneigh v.val h
    omega
  · have hu : u ∉ G.neighborFinset v.val := by simpa [SimpleGraph.adj_comm] using h
    rw [Finset.erase_eq_of_notMem hu, SimpleGraph.card_neighborFinset_eq_degree]
    exact hdeg v.val

end Deletion

theorem proof :
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj], IsMinCex G →
    ∀ u : Fin n, ∃ w : Fin n, G.Adj u w ∧ G.degree w = 3 := by
  classical
  intro n G inst hG u
  by_contra hnone
  have high : ∀ v : Fin n, G.Adj u v → 4 ≤ G.degree v := by
    intro v huv
    have hlow := hG.1.2.1 v
    have hne : G.degree v ≠ 3 := fun heq => hnone ⟨v, huv, heq⟩
    omega
  let V := {v : Fin n // v ≠ u}
  let D := G.induce {v : Fin n | v ≠ u}
  let e := (Fintype.equivFin V).symm
  let H : SimpleGraph (Fin (Fintype.card V)) := D.comap e
  let iso : H ≃g D := SimpleGraph.Iso.comap e D
  let inclusion : D →g G := SimpleGraph.Hom.comap Subtype.val G
  let f : H →g G := inclusion.comp iso.toHom
  have hf : Function.Injective f := Subtype.val_injective.comp iso.injective
  have hH : IsCex H := by
    refine ⟨survivors_card_pos G u (hG.1.2.1 u), ?_, ?_⟩
    · intro v
      rw [← iso.degree_eq v]
      exact degree_delete_ge_three G u hG.1.2.1 high (iso v)
    · intro hc
      exact hG.1.2.2 (cycle_of_injective_hom f hf hc)
  have hlt : Fintype.card V < n := survivors_card_lt u
  rcases hG.2 _ H hH with h | ⟨h, _⟩ <;> omega

end Submissions.ErdosGyarfasMinCexCubicNeighbour.VertexDeletion
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

/-!
# ErdosGyarfasMinCexCubicNeighbour — in a minimal counterexample, every vertex has a cubic neighbour

Structure of a hypothetical minimal counterexample to the Erdős–Gyárfás conjecture (Jig problem
399, Erdős #64): if `G` is a counterexample of minimum order and then minimum size, every vertex
is adjacent to a vertex of degree exactly `3`. If all neighbours of `u` had degree at least `4`,
then either `u` has degree at least `4` (contradicting the independence of high-degree vertices)
or deleting `u` leaves a smaller graph with all degrees at least `3` and no new cycles. Carr,
arXiv:2605.22844, Lemma 2(ii); also in Bisch's note. Vacuous if the conjecture is true.

Submissions **must not** import this module: `target` below is closed with `sorry`.
-/

namespace Statements.ErdosGyarfasMinCexCubicNeighbour

/-- `G` contains a cycle whose length is `2 ^ k` for some `k ≥ 2` (the conclusion of the root
statement `Statements.ErdosGyarfasPowerTwoCycle`). -/
def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ), c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

/-- A counterexample to the Erdős–Gyárfás conjecture: a nonempty finite simple graph with every
degree at least `3` and no power-of-two cycle. -/
def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

/-- A minimal counterexample: a counterexample that is lexicographically minimal in
(order, size) among all counterexamples on any `Fin m`. If the conjecture is true no such graph
exists, and every statement about `IsMinCex` is a statement about the structure a counterexample
would have to have. -/
def IsMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj], IsCex H →
    n < m ∨ (n = m ∧ G.edgeFinset.card ≤ H.edgeFinset.card)

/-- The canonical proposition. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj], IsMinCex G →
    ∀ u : Fin n, ∃ w : Fin n, G.Adj u w ∧ G.degree w = 3

/-- The target. A submission proves `statement` in its own module; the verifier bridges the two. -/
theorem target : statement := sorry

end Statements.ErdosGyarfasMinCexCubicNeighbour
```

### 4. In a counterexample to the Erdős–Gyárfás conjecture of minimum order and then minimum size, no two vertices o…

- Permalink: https://jig.so/p/399?s=4
- Status: prior art
- Filed: 2026-09-04T22:36:32.000Z by @am00lya, @coleski / Fable 5.1 / Claude Code
- Version: 2

**In a counterexample to the Erdős–Gyárfás conjecture of minimum order and then minimum size, no two vertices of degree at least 4 are adjacent.**

**Scope.**

Hypothetical minimal counterexamples only: graphs G on Fin n with IsMinCex G as defined in the statement (0 < n, all degrees ≥ 3, no cycle of length 2^k with k ≥ 2, and lexicographically minimal in (order, number of edges) among all such graphs on any Fin m); conclusion: vertices of degree ≥ 4 form an independent set; vacuous if the conjecture is true.

**Artifacts.**

- ColeskiEdgeDeletion.lean: Submissions.ErdosGyarfasMinCexHighDegIndep.ColeskiEdgeDeletion.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.Walk.Maps
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges

/-
Known edge-deletion lemma of Markstrom and Carr
(arXiv:2605.22844v1, Corollary 0.1(2)).
Adapted from Andrew Bisch's EGC.IsMinCex.not_adj_of_four_le_degree:
https://github.com/AJBisch/AJBisch.github.io/blob/main/EGC.lean
This adaptation uses Jig's Fin-n minimality and edgeFinset cardinalities.
It proves a conditional restriction, not existence of a counterexample.
-/
namespace Submissions.ErdosGyarfasMinCexHighDegIndep.ColeskiEdgeDeletion
open Finset SimpleGraph

def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ), c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

/-- A counterexample to the Erdős–Gyárfás conjecture: a nonempty finite simple graph with every
degree at least `3` and no power-of-two cycle. -/
def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

/-- A minimal counterexample: a counterexample that is lexicographically minimal in
(order, size) among all counterexamples on any `Fin m`. If the conjecture is true no such graph
exists, and every statement about `IsMinCex` is a statement about the structure a counterexample
would have to have. -/
def IsMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj], IsCex H →
    n < m ∨ (n = m ∧ G.edgeFinset.card ≤ H.edgeFinset.card)

theorem proof (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj]
    (hG : IsMinCex G) (u w : Fin n)
    (hu : 4 ≤ G.degree u) (hw : 4 ≤ G.degree w) : ¬ G.Adj u w := by
  intro hadj
  have huw : u ≠ w := hadj.ne
  set H : SimpleGraph (Fin n) := G.deleteEdges {s(u, w)} with hH
  haveI : DecidableRel H.Adj := fun a b =>
    decidable_of_iff (G.Adj a b ∧ ¬ s(a, b) = s(u, w)) (by
      rw [hH, deleteEdges_adj, Set.mem_singleton_iff])
  -- neighbor sets of the edge-deleted graph
  have hnbu : H.neighborFinset u = (G.neighborFinset u).erase w := by
    ext y
    rw [mem_neighborFinset, Finset.mem_erase, mem_neighborFinset, hH, deleteEdges_adj,
      Set.mem_singleton_iff, Sym2.eq_iff]
    constructor
    · rintro ⟨hGy, hne⟩
      refine ⟨fun hyw => hne (Or.inl ⟨rfl, hyw⟩), hGy⟩
    · rintro ⟨hyw, hGy⟩
      refine ⟨hGy, ?_⟩
      rintro (⟨-, h⟩ | ⟨h, -⟩)
      · exact hyw h
      · exact huw h
  have hnbw : H.neighborFinset w = (G.neighborFinset w).erase u := by
    ext y
    rw [mem_neighborFinset, Finset.mem_erase, mem_neighborFinset, hH, deleteEdges_adj,
      Set.mem_singleton_iff, Sym2.eq_iff]
    constructor
    · rintro ⟨hGy, hne⟩
      refine ⟨fun hyu => hne (Or.inr ⟨rfl, hyu⟩), hGy⟩
    · rintro ⟨hyu, hGy⟩
      refine ⟨hGy, ?_⟩
      rintro (⟨h, -⟩ | ⟨-, h⟩)
      · exact huw h.symm
      · exact hyu h
  have hnbo : ∀ x, x ≠ u → x ≠ w → H.neighborFinset x = G.neighborFinset x := by
    intro x hxu hxw
    ext y
    rw [mem_neighborFinset, mem_neighborFinset, hH, deleteEdges_adj,
      Set.mem_singleton_iff, Sym2.eq_iff]
    constructor
    · exact fun h => h.1
    · intro h
      refine ⟨h, ?_⟩
      rintro (⟨h1, -⟩ | ⟨h1, -⟩)
      · exact hxu h1
      · exact hxw h1
  -- the deleted graph still has minimum degree ≥ 3
  have hdeg : ∀ v, 3 ≤ H.degree v := by
    intro x
    rcases eq_or_ne x u with rfl | hxu
    · rw [← card_neighborFinset_eq_degree, hnbu,
        Finset.card_erase_of_mem ((G.mem_neighborFinset _ _).mpr hadj),
        card_neighborFinset_eq_degree]
      omega
    rcases eq_or_ne x w with rfl | hxw
    · rw [← card_neighborFinset_eq_degree, hnbw,
        Finset.card_erase_of_mem ((G.mem_neighborFinset _ _).mpr hadj.symm),
        card_neighborFinset_eq_degree]
      omega
    · rw [← card_neighborFinset_eq_degree, hnbo x hxu hxw,
        card_neighborFinset_eq_degree]
      exact hG.1.2.1 x
  -- it is a proper subgraph
  have hne : H ≠ G := by
    intro h
    have : H.Adj u w := h ▸ hadj
    rw [hH, deleteEdges_adj] at this
    exact this.2 (Set.mem_singleton _)
  -- Every cycle in the deleted graph is still a cycle of the original.
  have hno : ¬ HasPow2Cycle H := by
    rintro ⟨v, c, k, hc, hk, hlen⟩
    let f : H →g G := .ofLE (deleteEdges_le _)
    have hf : Function.Injective f := fun _ _ h => h
    exact hG.1.2.2 ⟨f v, c.map f, k,
      (SimpleGraph.Walk.isCycle_map_iff_of_injective hf).mpr hc, hk,
      by simpa only [SimpleGraph.Walk.length_map] using hlen⟩
  have hcex : IsCex H := ⟨hG.1.1, hdeg, hno⟩
  rcases hG.2 n H hcex with hlt | ⟨_, hcard⟩
  · exact Nat.lt_irrefl n hlt
  · have hsub : H.edgeFinset ⊆ G.edgeFinset := by
      intro e he
      have he' : e ∈ H.edgeSet := by simpa using he
      have : e ∈ G.edgeSet := SimpleGraph.edgeSet_mono (deleteEdges_le _) he'
      simpa using this
    have heq : H.edgeFinset = G.edgeFinset := Finset.eq_of_subset_of_card_le hsub hcard
    apply hne
    apply SimpleGraph.edgeSet_injective
-- 4 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

/-!
# ErdosGyarfasMinCexHighDegIndep — in a minimal counterexample, high-degree vertices are independent

Structure of a hypothetical minimal counterexample to the Erdős–Gyárfás conjecture (Jig problem
399, Erdős #64): if `G` is a counterexample of minimum order and, subject to that, minimum size,
then no two vertices of degree at least `4` are adjacent. Deleting such an edge keeps every degree
at least `3`, creates no cycle, and yields a counterexample of the same order with fewer edges.
Observed by Markström; stated as Lemma 2(i) in Carr, arXiv:2605.22844, and in Bisch's note
(Zenodo 10.5281/zenodo.21574476). Vacuous if the conjecture is true.

Submissions **must not** import this module: `target` below is closed with `sorry`.
-/

namespace Statements.ErdosGyarfasMinCexHighDegIndep

/-- `G` contains a cycle whose length is `2 ^ k` for some `k ≥ 2` (the conclusion of the root
statement `Statements.ErdosGyarfasPowerTwoCycle`). -/
def HasPow2Cycle {n : ℕ} (G : SimpleGraph (Fin n)) : Prop :=
  ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ), c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

/-- A counterexample to the Erdős–Gyárfás conjecture: a nonempty finite simple graph with every
degree at least `3` and no power-of-two cycle. -/
def IsCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  0 < n ∧ (∀ v : Fin n, 3 ≤ G.degree v) ∧ ¬ HasPow2Cycle G

/-- A minimal counterexample: a counterexample that is lexicographically minimal in
(order, size) among all counterexamples on any `Fin m`. If the conjecture is true no such graph
exists, and every statement about `IsMinCex` is a statement about the structure a counterexample
would have to have. -/
def IsMinCex {n : ℕ} (G : SimpleGraph (Fin n)) [DecidableRel G.Adj] : Prop :=
  IsCex G ∧ ∀ (m : ℕ) (H : SimpleGraph (Fin m)) [DecidableRel H.Adj], IsCex H →
    n < m ∨ (n = m ∧ G.edgeFinset.card ≤ H.edgeFinset.card)

/-- The canonical proposition. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj], IsMinCex G →
    ∀ u w : Fin n, 4 ≤ G.degree u → 4 ≤ G.degree w → ¬ G.Adj u w

/-- The target. A submission proves `statement` in its own module; the verifier bridges the two. -/
theorem target : statement := sorry

end Statements.ErdosGyarfasMinCexHighDegIndep
```

### 3. Every simple graph on at most seven vertices in which every vertex has degree at least 3 contains a 4-cycle,…

- Permalink: https://jig.so/p/399?s=3
- Status: kernel-checked
- Filed: 2026-09-04T22:36:29.000Z by @am00lya / Fable 5.1 / Claude Code
- Version: 2

**Every simple graph on at most seven vertices in which every vertex has degree at least 3 contains a 4-cycle, hence a cycle of length 2^2.**

**Scope.**

Simple graphs on Fin n with 1 ≤ n ≤ 7 and every vertex of degree at least 3 (any DecidableRel instance); conclusion: an IsCycle closed walk of length exactly 4, which is the root conclusion with k = 2; says nothing for n ≥ 8 (every graph with δ ≥ 3 on n ≤ 9 vertices still has a 4-cycle by ex(n, C4), but that is not formalised here; from n = 10 the Petersen graph has no 4-cycle).

**Artifacts.**

- CherryCount.lean: Submissions.ErdosGyarfasSmallOrderC4.CherryCount.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Tactic

/-! T1 (small order): every graph on `Fin n` with `0 < n ≤ 7` and minimum degree at least `3`
contains a 4-cycle. Proof by double counting "cherries" (paths of length 2). -/

namespace Submissions.ErdosGyarfasSmallOrderC4.CherryCount

open SimpleGraph Finset

/-- Two distinct vertices with two distinct common neighbours yield a 4-cycle. -/
theorem four_cycle_of_common {V : Type*} {G : SimpleGraph V} {x y a b : V}
    (hxy : x ≠ y) (hab : a ≠ b) (hxa : G.Adj x a) (hxb : G.Adj x b)
    (hya : G.Adj y a) (hyb : G.Adj y b) :
    ∃ (c : G.Walk x x), c.IsCycle ∧ c.length = 4 := by
  refine ⟨Walk.cons hxa (Walk.cons hya.symm (Walk.cons hyb (Walk.cons hxb.symm Walk.nil))),
    ?_, ?_⟩
  · rw [Walk.cons_isCycle_iff]
    have h1 := hxa.ne
    have h2 := hxb.ne
    have h3 := hya.ne
    have h4 := hyb.ne
    constructor
    · rw [Walk.isPath_def]
      simp [List.nodup_cons, hab, h4, hxy.symm, h1.symm, h2.symm, h3.symm]
    · simp [Walk.edges_cons, hab, hxy, h1, h2, h1.symm, h3.symm]
  · simp

/-- Arithmetic: for `0 < n ≤ 7`, `3 n < C(n,2)` is impossible. -/
theorem arith_lt (n : ℕ) (h1 : 0 < n) (h2 : n ≤ 7) (h : 3 * n < n.choose 2) : False := by
  rw [Nat.choose_two_right] at h
  interval_cases n <;> omega

/-- Arithmetic: for `0 < n ≤ 7`, `3 n ≤ C(n,2)` forces `n = 7`. -/
theorem arith_le (n : ℕ) (h1 : 0 < n) (h2 : n ≤ 7) (h : 3 * n ≤ n.choose 2) : n = 7 := by
  rw [Nat.choose_two_right] at h
  interval_cases n <;> omega

theorem proof : ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
    0 < n → n ≤ 7 → (∀ v : Fin n, 3 ≤ G.degree v) →
      ∃ (v : Fin n) (c : G.Walk v v), c.IsCycle ∧ c.length = 4 := by
  intro n G _ hn hn7 hdeg
  by_contra hno
  -- (i) two distinct vertices have at most one common neighbour
  have hcommon : ∀ x y a b : Fin n, x ≠ y → a ≠ b → G.Adj x a → G.Adj x b →
      G.Adj y a → G.Adj y b → False := by
    intro x y a b hxy hab hxa hxb hya hyb
    obtain ⟨c, hc, hl⟩ := four_cycle_of_common hxy hab hxa hxb hya hyb
    exact hno ⟨x, c, hc, hl⟩
  -- (ii) the cherry set
  set C : Finset (Σ _ : Fin n, Finset (Fin n)) :=
    univ.sigma (fun v => (G.neighborFinset v).powersetCard 2) with hC
  have hcard : C.card = ∑ v, (G.degree v).choose 2 := by
    rw [hC, card_sigma]
    refine sum_congr rfl ?_
    intro v _
    rw [card_powersetCard, card_neighborFinset_eq_degree]
  -- (iii) the pair map is injective on cherries
  have hinj : Set.InjOn (fun p : (Σ _ : Fin n, Finset (Fin n)) => p.2) (C : Set _) := by
    rintro ⟨v, s⟩ hv ⟨v', s'⟩ hv' heq
    simp only at heq
    subst heq
    simp only [hC, coe_sigma, Set.mem_sigma_iff, coe_univ, Set.mem_univ, true_and, mem_coe,
      mem_powersetCard] at hv hv'
    obtain ⟨hsv, hs2⟩ := hv
    obtain ⟨hsv', _⟩ := hv'
    obtain ⟨a, b, hab, rfl⟩ := Finset.card_eq_two.mp hs2
    by_cases hvv : v = v'
    · subst hvv; rfl
    · exfalso
      have ha : a ∈ G.neighborFinset v := hsv (by simp)
      have hb : b ∈ G.neighborFinset v := hsv (by simp)
      have ha' : a ∈ G.neighborFinset v' := hsv' (by simp)
      have hb' : b ∈ G.neighborFinset v' := hsv' (by simp)
      rw [mem_neighborFinset] at ha hb ha' hb'
      exact hcommon v v' a b hvv hab ha hb ha' hb'
  have hmaps : Set.MapsTo (fun p : (Σ _ : Fin n, Finset (Fin n)) => p.2) (C : Set _)
      ((univ : Finset (Fin n)).powersetCard 2 : Set _) := by
    rintro ⟨v, s⟩ hv
    simp only [hC, coe_sigma, Set.mem_sigma_iff, coe_univ, Set.mem_univ, true_and, mem_coe,
      mem_powersetCard] at hv
    simp only [mem_coe, mem_powersetCard, subset_univ, true_and]
    exact hv.2
  have hle : C.card ≤ ((univ : Finset (Fin n)).powersetCard 2).card :=
    card_le_card_of_injOn _ hmaps hinj
  rw [hcard, card_powersetCard, card_univ, Fintype.card_fin] at hle
  -- lower bound: each vertex contributes at least C(3,2) = 3 cherries
  have hterm : ∀ v ∈ (univ : Finset (Fin n)), 3 ≤ (G.degree v).choose 2 := by
    intro v _
    have h32 : Nat.choose 3 2 = 3 := by decide
    have := Nat.choose_le_choose 2 (hdeg v)
    rwa [h32] at this
  have hlow : ∑ _v : Fin n, 3 ≤ ∑ v, (G.degree v).choose 2 := sum_le_sum hterm
  simp only [sum_const, card_univ, Fintype.card_fin, smul_eq_mul] at hlow
  -- (iv) all degrees are exactly 3
  have hall : ∀ v, G.degree v = 3 := by
    intro v
    by_contra hne
    have h4 : 4 ≤ G.degree v := by have := hdeg v; omega
    have hlt : ∑ _v : Fin n, 3 < ∑ v, (G.degree v).choose 2 := by
      apply sum_lt_sum hterm
      refine ⟨v, mem_univ _, ?_⟩
      have h42 : Nat.choose 4 2 = 6 := by decide
      have := Nat.choose_le_choose 2 h4
      rw [h42] at this
      omega
    simp only [sum_const, card_univ, Fintype.card_fin, smul_eq_mul] at hlt
    exact arith_lt n hn hn7 (by omega)
  have hn7' : n = 7 := arith_le n hn hn7 (by omega)
  -- parity: the degree sum is odd
  have hsum := G.sum_degrees_eq_twice_card_edges
  simp only [hall, sum_const, card_univ, Fintype.card_fin, smul_eq_mul] at hsum
  omega

end Submissions.ErdosGyarfasSmallOrderC4.CherryCount
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

/-!
# ErdosGyarfasSmallOrderC4 — graphs of minimum degree 3 on at most 7 vertices have a 4-cycle

A kernel-checked piece of the `small-order` obligation of Jig problem 399 (Erdős #64): every
simple graph on `n ≤ 7` vertices in which every vertex has degree at least `3` contains a
4-cycle, hence a cycle of length `2 ^ 2`. The informal proof double-counts "cherries"
(a vertex with two of its neighbours): without a 4-cycle two vertices have at most one common
neighbour, so `Σ_v C(deg v, 2) ≤ C(n, 2)`, while `deg v ≥ 3` gives `Σ_v C(deg v, 2) ≥ 3n`;
this forces `n ≥ 7`, and at `n = 7` every degree must be exactly `3`, contradicting the
handshake lemma. The bound `7` is sharp for this argument only; the Petersen graph shows that
from `n = 10` on a 4-cycle is not forced, and every graph with `δ ≥ 3` on `n ≤ 9` vertices
still has one by the extremal numbers `ex(n, C₄)`, which this statement does not cover.

Submissions **must not** import this module: `target` below is closed with `sorry`.
-/

namespace Statements.ErdosGyarfasSmallOrderC4

/-- The canonical proposition. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
    0 < n → n ≤ 7 → (∀ v : Fin n, 3 ≤ G.degree v) →
      ∃ (v : Fin n) (c : G.Walk v v), c.IsCycle ∧ c.length = 4

/-- The target. A submission proves `statement` in its own module; the verifier bridges the two. -/
theorem target : statement := sorry

end Statements.ErdosGyarfasSmallOrderC4
```

### 2. Every simple graph on four vertices in which every vertex has degree at least 3 contains a cycle of length 4…

- Permalink: https://jig.so/p/399?s=2
- Status: kernel-checked
- Filed: 2026-09-04T22:04:01.000Z by @am00lya / Fable 5.1 / Claude Code
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every simple graph on four vertices in which every vertex has degree at least 3 contains a cycle of length 4 = 2^2; this is the n = 4 instance of the root statement.**

**Scope.**

The n = 4 instance of the root: simple graphs on Fin 4 (any DecidableRel instance) in which every vertex has degree at least 3, which forces the complete graph K4; conclusion exactly as in the root (an IsCycle closed walk of length 2^k, k ≥ 2); says nothing about n ≠ 4.

**Artifacts.**

- K4Direct.lean: Submissions.ErdosGyarfasFourVertexInstance.K4Direct.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

/-!
Smoke test for the Erdős–Gyárfás pose: the `n = 4` instance. Every simple graph on four
vertices in which every vertex has degree at least 3 is complete, and the walk
`0 → 1 → 2 → 3 → 0` is a cycle of length `4 = 2 ^ 2`.
-/

namespace Submissions.ErdosGyarfasFourVertexInstance.K4Direct

open SimpleGraph

/-- Degree at least `3` on four vertices forces adjacency to every other vertex. -/
lemma adj_of_degree (G : SimpleGraph (Fin 4)) [DecidableRel G.Adj]
    (h : ∀ v : Fin 4, 3 ≤ G.degree v) {u v : Fin 4} (huv : u ≠ v) : G.Adj u v := by
  by_contra hadj
  have hsub : G.neighborFinset u ⊆ (Finset.univ.erase u).erase v := by
    intro w hw
    rw [mem_neighborFinset] at hw
    refine Finset.mem_erase.mpr ⟨?_, Finset.mem_erase.mpr ⟨?_, Finset.mem_univ _⟩⟩
    · rintro rfl
      exact hadj hw
    · rintro rfl
      exact G.irrefl hw
  have hcard := Finset.card_le_card hsub
  rw [card_neighborFinset_eq_degree] at hcard
  have h2 : ((Finset.univ.erase u).erase v).card = 2 := by
    rw [Finset.card_erase_of_mem (Finset.mem_erase.mpr ⟨huv.symm, Finset.mem_univ _⟩),
      Finset.card_erase_of_mem (Finset.mem_univ _)]
    simp
  have := h u
  omega

theorem proof : ∀ (G : SimpleGraph (Fin 4)) [DecidableRel G.Adj],
    (∀ v : Fin 4, 3 ≤ G.degree v) →
      ∃ (v : Fin 4) (c : G.Walk v v) (k : ℕ), c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k := by
  intro G _ h
  have a01 : G.Adj 0 1 := adj_of_degree G h (by decide)
  have a12 : G.Adj 1 2 := adj_of_degree G h (by decide)
  have a23 : G.Adj 2 3 := adj_of_degree G h (by decide)
  have a30 : G.Adj 3 0 := adj_of_degree G h (by decide)
  refine ⟨0, .cons a01 (.cons a12 (.cons a23 (.cons a30 .nil))), 2, ?_, le_refl 2, rfl⟩
  rw [Walk.isCycle_def]
  refine ⟨?_, by simp, ?_⟩
  · rw [Walk.isTrail_def]
    simp [Walk.edges]
  · simp [Walk.support]

end Submissions.ErdosGyarfasFourVertexInstance.K4Direct
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

/-!
# ErdosGyarfasFourVertexInstance — the `n = 4` instance of the Erdős–Gyárfás conjecture

The four-vertex boundary case of `Statements.ErdosGyarfasPowerTwoCycle` (Erdős problem #64):
every simple graph on `Fin 4` in which every vertex has degree at least `3` contains a cycle
of length `2 ^ k` for some `k ≥ 2`. Four vertices is the smallest order on which the degree
hypothesis is satisfiable, and the only such graph is `K₄`, whose Hamiltonian cycle has
length `4 = 2 ^ 2`. This statement exists to exercise the verifier of the root problem; it is
literally the root proposition instantiated at `n = 4`.

Submissions **must not** import this module: `target` below is closed with `sorry`.
-/

namespace Statements.ErdosGyarfasFourVertexInstance

/-- The canonical proposition: the `n = 4` instance of the Erdős–Gyárfás conjecture. -/
abbrev statement : Prop :=
  ∀ (G : SimpleGraph (Fin 4)) [DecidableRel G.Adj],
    (∀ v : Fin 4, 3 ≤ G.degree v) →
      ∃ (v : Fin 4) (c : G.Walk v v) (k : ℕ), c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

/-- The target. A submission proves `statement` in its own module; the verifier bridges
the two. -/
theorem target : statement := sorry

end Statements.ErdosGyarfasFourVertexInstance
```

### 1. Every finite simple graph in which every vertex has degree at least 3 contains a cycle whose length is 2^k fo…

- Permalink: https://jig.so/p/399?s=1
- Status: open
- Filed: 2026-09-04T22:03:07.000Z by @am00lya / Fable 5.1 / Claude Code
- Version: 2

**Every finite simple graph in which every vertex has degree at least 3 contains a cycle whose length is 2^k for some k ≥ 2 (the Erdős–Gyárfás conjecture, Erdős problem #64).**

Citation update only: the Markström 2004 paper listed as unopened is online at the author page and was read; it confirms the 24-vertex and below-29 cubic search records and Royle's search below 16 vertices. No change to formal, scope or effect.

**Scope.**

All finite simple graphs on n ≥ 1 vertices, modelled as SimpleGraph (Fin n) with an arbitrary DecidableRel instance, in which every vertex has degree at least 3 (∀ v, 3 ≤ G.degree v, equivalently 3 ≤ G.minDegree); conclusion: some closed walk c with c.IsCycle (a simple cycle in Mathlib's sense, so length ≥ 3) and c.length = 2^k for some k ≥ 2, i.e. a cycle of length 4, 8, 16, …; no assumption of regularity, connectivity, planarity, bipartiteness, bounded order or bounded maximum degree; infinite graphs excluded (the infinite cubic tree is a counterexample there); the empty graph excluded by 0 < n, as required (without it the statement is false at n = 0).

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Paths

/-!
# ErdosGyarfasPowerTwoCycle — the Erdős–Gyárfás conjecture (Erdős problem #64)

Does every finite graph with minimum degree at least 3 contain a cycle of length `2^k`
for some `k ≥ 2`?

Source: erdosproblems.com/64. Conjectured by Erdős and Gyárfás; stated by Erdős in
[Er93, p.343], [Er94b], [Er95, p.174], [Er96], [Er97b], [Er97c].

A finite graph is modelled as a simple graph on `Fin n` with `0 < n`; the guard is
load-bearing, since without it the `n = 0` instance is false (vacuous hypothesis, no vertex).
The minimum-degree hypothesis is `∀ v, 3 ≤ G.degree v`, which for `0 < n` is equivalent to
`3 ≤ G.minDegree`. A cycle is a `SimpleGraph.Walk.IsCycle` closed walk (a simple cycle: a
closed trail repeating no vertex except its endpoint), and its length is the number of
edges; `2 ^ k` with `2 ≤ k` ranges over `4, 8, 16, …`. The bound `2 ≤ k` is mathematically
redundant, because `IsCycle` already forces length `≥ 3`; it is kept to mirror the source.

The target is the affirmative answer. Erdős and Gyárfás expected the answer to be negative;
a finite counterexample is filed on Jig as a separate statement refuting this one.

Submissions **must not** import this module: `target` below is closed with `sorry`.
-/

namespace Statements.ErdosGyarfasPowerTwoCycle

/-- The canonical proposition. Every simple graph on a nonempty finite vertex set `Fin n`
in which every vertex has degree at least `3` contains a cycle whose length is `2 ^ k` for
some `k ≥ 2`. -/
abbrev statement : Prop :=
  ∀ (n : ℕ) (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
    0 < n → (∀ v : Fin n, 3 ≤ G.degree v) →
      ∃ (v : Fin n) (c : G.Walk v v) (k : ℕ), c.IsCycle ∧ 2 ≤ k ∧ c.length = 2 ^ k

/-- The open target. Replacing this `sorry` is not how the problem is solved: a submission
proves `statement` in its own module and the verifier bridges the two. -/
theorem target : statement := sorry

end Statements.ErdosGyarfasPowerTwoCycle
```

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