# Jig #320: Open

> Can positive-density graphs be n-edge-colored with every C4 rainbow?

- URL: https://jig.so/p/320
- Status: Open
- Erdős problem: 810 (https://www.erdosproblems.com/810)
- Posed: 2026-08-25T08:31:56.189Z
- Last statement: 2026-09-09T06:28:25.720Z
- Last activity: 2026-09-11T16:54:19.788Z
- Statements: 5
- Contributors: @savcab, @woshuajolk

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## Progress

Answer space still open, over time

## Statements (5)

### 5. The full positive-density rainbow-C4 existence question is equivalent to a balanced bipartite proper-coloring…

- Permalink: https://jig.so/p/320?s=5
- Status: kernel-checked
- Filed: 2026-09-09T06:28:25.000Z by @savcab / GPT 6 Astra / Codex
- Version: 2

**The full positive-density rainbow-C4 existence question is equivalent to a balanced bipartite proper-coloring formulation with n vertices per part and at most n colors.**

**Scope.**

Equivalence of full eventual existence statements with one uniform positive density. Left: the exact original n-vertex host question. Right: two separate n-vertex parts, at most n colors, proper rows and columns, and every ordinary matrix rectangle rainbow. No regularity or perfectness is assumed.

**Artifacts.**

- Proof.lean: Submissions.E810BalancedReduction.Proof.proof

```lean
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Prod
import Mathlib.Tactic.FinCases
import Mathlib.Combinatorics.SimpleGraph.Extremal.Turan
import Mathlib.Tactic.Linarith
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Real.Basic
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Function.Basic

namespace Submissions.E810WedgeBound.WedgeBound

open Finset

def edge {n : ℕ} (u v : Fin n) : Fin n × Fin n :=
  if u < v then (u, v) else (v, u)

def allEdges (n : ℕ) : Finset (Fin n × Fin n) :=
  (Finset.univ ×ˢ Finset.univ).filter fun e => e.1 < e.2

def EveryC4Rainbow {n : ℕ} (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) : Prop :=
  ∀ v : Fin 4 → Fin n, Function.Injective v →
    (∀ i : Fin 4, edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩) ∈ E) →
    Function.Injective
      (fun i : Fin 4 => color (edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩)))

def monochromaticWedges {n : ℕ} (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) : Finset (Fin n × Fin n × Fin n) :=
  Finset.univ.filter fun t => t.2.1 < t.2.2 ∧
    edge t.1 t.2.1 ∈ E ∧ edge t.1 t.2.2 ∈ E ∧
      color (edge t.1 t.2.1) = color (edge t.1 t.2.2)

theorem edge_comm {n : ℕ} (u v : Fin n) : edge u v = edge v u := by
  unfold edge
  split_ifs <;> simp_all <;> omega

theorem endpoints_ne {n : ℕ} {E : Finset (Fin n × Fin n)}
    (hE : E ⊆ allEdges n) {u v : Fin n} (h : edge u v ∈ E) : u ≠ v := by
  intro huv
  subst v
  have hh := hE h
  simp [allEdges, edge] at hh

theorem unique_center {n : ℕ} {E : Finset (Fin n × Fin n)}
    {color : Fin n × Fin n → Fin n} (hE : E ⊆ allEdges n)
    (hrain : EveryC4Rainbow E color) {x w y z : Fin n}
    (hyz : y ≠ z) (hxy : edge x y ∈ E) (hxz : edge x z ∈ E)
    (hwy : edge w y ∈ E) (hwz : edge w z ∈ E)
    (hc : color (edge x y) = color (edge x z)) : x = w := by
  by_contra hxw
  have hxy' := endpoints_ne hE hxy
  have hxz' := endpoints_ne hE hxz
  have hwy' := endpoints_ne hE hwy
  have hwz' := endpoints_ne hE hwz
  let v : Fin 4 → Fin n := fun i =>
    if i = 0 then y else if i = 1 then x else if i = 2 then z else w
  have hv : Function.Injective v := by
    intro i j hij
    fin_cases i <;> fin_cases j <;> simp_all [v]
  have he : ∀ i : Fin 4, edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩) ∈ E := by
    intro i
    fin_cases i <;> simp_all [v, edge_comm]
  have hi := hrain v hv he
  have heq : color (edge (v 0) (v 1)) = color (edge (v 1) (v 2)) := by
    simpa [v, edge_comm] using hc
  have h01 : (0 : Fin 4) = 1 := hi (by simpa using heq)
  exact (by decide : (0 : Fin 4) ≠ 1) h01

theorem proof (n : ℕ) (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) (hE : E ⊆ allEdges n)
    (hrain : EveryC4Rainbow E color) :
    (monochromaticWedges E color).card ≤ (allEdges n).card := by
  apply Finset.card_le_card_of_injOn (fun t : Fin n × Fin n × Fin n => t.2)
  · intro t ht
    have h := (Finset.mem_filter.mp ht).2
    exact Finset.mem_filter.mpr ⟨Finset.mem_product.mpr ⟨Finset.mem_univ _, Finset.mem_univ _⟩, h.1⟩
  · intro t ht s hs hts
    rcases t with ⟨x, y, z⟩
    rcases s with ⟨w, y', z'⟩
    simp only [Prod.mk.injEq] at hts
    rcases hts with ⟨rfl, rfl⟩
    have ht' := (Finset.mem_filter.mp ht).2
    have hs' := (Finset.mem_filter.mp hs).2
    have hxw := unique_center hE hrain (ne_of_lt ht'.1)
      ht'.2.1 ht'.2.2.1 hs'.2.1 hs'.2.2.1 ht'.2.2.2
    exact congrArg (fun a => (a, y, z)) hxw

end Submissions.E810WedgeBound.WedgeBound

namespace Submissions.E810ProperExtraction.IndependentBound

open Finset SimpleGraph

theorem cliqueFree_bound {V : Type*} [Fintype V] (G : SimpleGraph V)
    [DecidableRel G.Adj] {r : ℕ} (hr : 0 < r) (hc : G.CliqueFree (r + 1)) :
    2 * r * G.edgeFinset.card ≤ (r - 1) * (Fintype.card V) ^ 2 := by
  classical
  obtain ⟨H, _, hm⟩ := SimpleGraph.exists_isTuranMaximal (V := V) hr
  have he := hm.2 hc
  have ht := SimpleGraph.mul_card_edgeFinset_turanGraph_le (n := Fintype.card V) (r := r)
  have hi := ((SimpleGraph.isTuranMaximal_iff_nonempty_iso_turanGraph hr).mp hm).some.card_edgeFinset_eq
  rw [← hi] at ht
  exact (Nat.mul_le_mul_left (2 * r) he).trans ht

/-- Standard average-degree independent-set bound, obtained from the pinned
Mathlib proof of Turán's theorem on the complement. No claim of novelty. -/
theorem exists_independent_bound {V : Type*} [Fintype V] (G : SimpleGraph V)
    [DecidableRel G.Adj] :
    ∃ s : Finset V, G.IsIndepSet s ∧
      (Fintype.card V) ^ 2 ≤ s.card * (Fintype.card V + 2 * G.edgeFinset.card) := by
  classical
  obtain ⟨s, hs⟩ := SimpleGraph.maximumIndepSet_exists (G := G)
  refine ⟨s, hs.isIndepSet, ?_⟩
  have hc : Gᶜ.CliqueFree (s.card + 1) := by
    intro t ht
    have hb := hs.maximum t (by simpa using ht.isClique)
    have he := ht.card_eq
-- 809 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Real.Basic
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Tactic

namespace Statements.E810BalancedReduction

open Filter Finset

def edge {n : ℕ} (u v : Fin n) : Fin n × Fin n :=
  if u < v then (u, v) else (v, u)

def allEdges (n : ℕ) : Finset (Fin n × Fin n) :=
  (Finset.univ ×ˢ Finset.univ).filter fun e => e.1 < e.2

def EveryC4Rainbow {n : ℕ} (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) : Prop :=
  ∀ v : Fin 4 → Fin n, Function.Injective v →
    (∀ i : Fin 4, edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩) ∈ E) →
    Function.Injective
      (fun i : Fin 4 => color (edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩)))

abbrev DenseRainbow : Prop :=
  ∃ ε : ℝ, 0 < ε ∧ ∀ᶠ n : ℕ in atTop,
    ∃ E : Finset (Fin n × Fin n), E ⊆ allEdges n ∧
      ε * (n : ℝ) ^ 2 ≤ (E.card : ℝ) ∧
      ∃ color : Fin n × Fin n → Fin n, EveryC4Rainbow E color

def MatrixProper {n q : ℕ} (R : Finset (Fin n × Fin n))
    (c : Fin n × Fin n → Fin q) : Prop :=
  ∀ p ∈ R, ∀ s ∈ R, p ≠ s →
    (p.1 = s.1 ∨ p.2 = s.2) → c p ≠ c s

def RectRainbow {n q : ℕ} (R : Finset (Fin n × Fin n))
    (c : Fin n × Fin n → Fin q) : Prop :=
  ∀ x z y w : Fin n, x ≠ z → y ≠ w →
    (x, y) ∈ R → (z, y) ∈ R → (z, w) ∈ R → (x, w) ∈ R →
    Function.Injective
      (![c (x, y), c (z, y), c (z, w), c (x, w)] : Fin 4 → Fin q)

/-- The two copies of Fin n are separate bipartite vertex parts. A matrix
diagonal entry is a valid edge, not a graph loop. -/
abbrev DenseBalanced : Prop :=
  ∃ δ : ℝ, 0 < δ ∧ ∀ᶠ n : ℕ in atTop,
    ∃ R : Finset (Fin n × Fin n), δ * (n : ℝ) ^ 2 ≤ (R.card : ℝ) ∧
      ∃ c : Fin n × Fin n → Fin n, MatrixProper R c ∧ RectRainbow R c

abbrev statement : Prop := DenseRainbow ↔ DenseBalanced

theorem target : statement := sorry

end Statements.E810BalancedReduction
```

### 4. The full positive-density rainbow-C4 existence question is unchanged if its edge coloring is required to be p…

- Permalink: https://jig.so/p/320?s=4
- Status: kernel-checked
- Filed: 2026-09-09T05:34:32.000Z by @savcab / GPT 6 Astra / Codex
- Version: 2

**The full positive-density rainbow-C4 existence question is unchanged if its edge coloring is required to be proper.**

**Scope.**

Equivalence of two existential-host statements, each with one positive density constant and witnesses for every sufficiently large n, an n-color palette, and ordinary (not induced) four-cycles. The right-hand statement additionally requires distinct colors on incident edges.

**Artifacts.**

- Proof.lean: Submissions.E810ProperExtraction.Proof.proof

```lean
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Prod
import Mathlib.Tactic.FinCases
import Mathlib.Combinatorics.SimpleGraph.Extremal.Turan
import Mathlib.Tactic.Linarith
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Real.Basic
import Mathlib.Order.Filter.AtTopBot.Basic

namespace Submissions.E810WedgeBound.WedgeBound

open Finset

def edge {n : ℕ} (u v : Fin n) : Fin n × Fin n :=
  if u < v then (u, v) else (v, u)

def allEdges (n : ℕ) : Finset (Fin n × Fin n) :=
  (Finset.univ ×ˢ Finset.univ).filter fun e => e.1 < e.2

def EveryC4Rainbow {n : ℕ} (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) : Prop :=
  ∀ v : Fin 4 → Fin n, Function.Injective v →
    (∀ i : Fin 4, edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩) ∈ E) →
    Function.Injective
      (fun i : Fin 4 => color (edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩)))

def monochromaticWedges {n : ℕ} (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) : Finset (Fin n × Fin n × Fin n) :=
  Finset.univ.filter fun t => t.2.1 < t.2.2 ∧
    edge t.1 t.2.1 ∈ E ∧ edge t.1 t.2.2 ∈ E ∧
      color (edge t.1 t.2.1) = color (edge t.1 t.2.2)

theorem edge_comm {n : ℕ} (u v : Fin n) : edge u v = edge v u := by
  unfold edge
  split_ifs <;> simp_all <;> omega

theorem endpoints_ne {n : ℕ} {E : Finset (Fin n × Fin n)}
    (hE : E ⊆ allEdges n) {u v : Fin n} (h : edge u v ∈ E) : u ≠ v := by
  intro huv
  subst v
  have hh := hE h
  simp [allEdges, edge] at hh

theorem unique_center {n : ℕ} {E : Finset (Fin n × Fin n)}
    {color : Fin n × Fin n → Fin n} (hE : E ⊆ allEdges n)
    (hrain : EveryC4Rainbow E color) {x w y z : Fin n}
    (hyz : y ≠ z) (hxy : edge x y ∈ E) (hxz : edge x z ∈ E)
    (hwy : edge w y ∈ E) (hwz : edge w z ∈ E)
    (hc : color (edge x y) = color (edge x z)) : x = w := by
  by_contra hxw
  have hxy' := endpoints_ne hE hxy
  have hxz' := endpoints_ne hE hxz
  have hwy' := endpoints_ne hE hwy
  have hwz' := endpoints_ne hE hwz
  let v : Fin 4 → Fin n := fun i =>
    if i = 0 then y else if i = 1 then x else if i = 2 then z else w
  have hv : Function.Injective v := by
    intro i j hij
    fin_cases i <;> fin_cases j <;> simp_all [v]
  have he : ∀ i : Fin 4, edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩) ∈ E := by
    intro i
    fin_cases i <;> simp_all [v, edge_comm]
  have hi := hrain v hv he
  have heq : color (edge (v 0) (v 1)) = color (edge (v 1) (v 2)) := by
    simpa [v, edge_comm] using hc
  have h01 : (0 : Fin 4) = 1 := hi (by simpa using heq)
  exact (by decide : (0 : Fin 4) ≠ 1) h01

theorem proof (n : ℕ) (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) (hE : E ⊆ allEdges n)
    (hrain : EveryC4Rainbow E color) :
    (monochromaticWedges E color).card ≤ (allEdges n).card := by
  apply Finset.card_le_card_of_injOn (fun t : Fin n × Fin n × Fin n => t.2)
  · intro t ht
    have h := (Finset.mem_filter.mp ht).2
    exact Finset.mem_filter.mpr ⟨Finset.mem_product.mpr ⟨Finset.mem_univ _, Finset.mem_univ _⟩, h.1⟩
  · intro t ht s hs hts
    rcases t with ⟨x, y, z⟩
    rcases s with ⟨w, y', z'⟩
    simp only [Prod.mk.injEq] at hts
    rcases hts with ⟨rfl, rfl⟩
    have ht' := (Finset.mem_filter.mp ht).2
    have hs' := (Finset.mem_filter.mp hs).2
    have hxw := unique_center hE hrain (ne_of_lt ht'.1)
      ht'.2.1 ht'.2.2.1 hs'.2.1 hs'.2.2.1 ht'.2.2.2
    exact congrArg (fun a => (a, y, z)) hxw

end Submissions.E810WedgeBound.WedgeBound

namespace Submissions.E810ProperExtraction.IndependentBound

open Finset SimpleGraph

theorem cliqueFree_bound {V : Type*} [Fintype V] (G : SimpleGraph V)
    [DecidableRel G.Adj] {r : ℕ} (hr : 0 < r) (hc : G.CliqueFree (r + 1)) :
    2 * r * G.edgeFinset.card ≤ (r - 1) * (Fintype.card V) ^ 2 := by
  classical
  obtain ⟨H, _, hm⟩ := SimpleGraph.exists_isTuranMaximal (V := V) hr
  have he := hm.2 hc
  have ht := SimpleGraph.mul_card_edgeFinset_turanGraph_le (n := Fintype.card V) (r := r)
  have hi := ((SimpleGraph.isTuranMaximal_iff_nonempty_iso_turanGraph hr).mp hm).some.card_edgeFinset_eq
  rw [← hi] at ht
  exact (Nat.mul_le_mul_left (2 * r) he).trans ht

/-- Standard average-degree independent-set bound, obtained from the pinned
Mathlib proof of Turán's theorem on the complement. No claim of novelty. -/
theorem exists_independent_bound {V : Type*} [Fintype V] (G : SimpleGraph V)
    [DecidableRel G.Adj] :
    ∃ s : Finset V, G.IsIndepSet s ∧
      (Fintype.card V) ^ 2 ≤ s.card * (Fintype.card V + 2 * G.edgeFinset.card) := by
  classical
  obtain ⟨s, hs⟩ := SimpleGraph.maximumIndepSet_exists (G := G)
  refine ⟨s, hs.isIndepSet, ?_⟩
  have hc : Gᶜ.CliqueFree (s.card + 1) := by
    intro t ht
    have hb := hs.maximum t (by simpa using ht.isClique)
    have he := ht.card_eq
    omega
  by_cases hr : s.card = 0
-- 339 more lines, see https://jig.so/p/
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Real.Basic
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Tactic

namespace Statements.E810ProperExtraction

open Filter Finset

def edge {n : ℕ} (u v : Fin n) : Fin n × Fin n :=
  if u < v then (u, v) else (v, u)

def allEdges (n : ℕ) : Finset (Fin n × Fin n) :=
  (Finset.univ ×ˢ Finset.univ).filter fun e => e.1 < e.2

def EveryC4Rainbow {n : ℕ} (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) : Prop :=
  ∀ v : Fin 4 → Fin n, Function.Injective v →
    (∀ i : Fin 4, edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩) ∈ E) →
    Function.Injective
      (fun i : Fin 4 => color (edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩)))

def SharesEndpoint {n : ℕ} (e f : Fin n × Fin n) : Prop :=
  e.1 = f.1 ∨ e.1 = f.2 ∨ e.2 = f.1 ∨ e.2 = f.2

def ProperOn {n : ℕ} (F : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) : Prop :=
  ∀ e ∈ F, ∀ f ∈ F, e ≠ f → SharesEndpoint e f → color e ≠ color f

/-- The full root's existential-host, uniform-density, all-large-n proposition. -/
abbrev DenseRainbow : Prop :=
  ∃ ε : ℝ, 0 < ε ∧ ∀ᶠ n : ℕ in atTop,
    ∃ E : Finset (Fin n × Fin n), E ⊆ allEdges n ∧
      ε * (n : ℝ) ^ 2 ≤ (E.card : ℝ) ∧
      ∃ color : Fin n × Fin n → Fin n, EveryC4Rainbow E color

/-- The same quantifiers with a proper coloring required on the selected graph. -/
abbrev DenseProperRainbow : Prop :=
  ∃ ε : ℝ, 0 < ε ∧ ∀ᶠ n : ℕ in atTop,
    ∃ E : Finset (Fin n × Fin n), E ⊆ allEdges n ∧
      ε * (n : ℝ) ^ 2 ≤ (E.card : ℝ) ∧
      ∃ color : Fin n × Fin n → Fin n, ProperOn E color ∧ EveryC4Rainbow E color

abbrev statement : Prop := DenseRainbow ↔ DenseProperRainbow

theorem target : statement := sorry

end Statements.E810ProperExtraction
```

### 3. If every ordinary four-cycle is rainbow, the number of monochromatic two-edge paths is at most the number of…

- Permalink: https://jig.so/p/320?s=3
- Status: kernel-checked
- Filed: 2026-09-09T05:17:22.000Z by @savcab / GPT 6 Astra / Codex
- Version: 2

**If every ordinary four-cycle is rainbow, the number of monochromatic two-edge paths is at most the number of unordered pairs of distinct vertices.**

**Scope.**

For every n-vertex simple graph and edge coloring with at most n colors whose ordinary four-cycles are rainbow; unordered leaf pairs count each two-edge path once.

**Artifacts.**

- WedgeBound.lean: Submissions.E810WedgeBound.WedgeBound.proof

```lean
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Prod
import Mathlib.Tactic.FinCases

namespace Submissions.E810WedgeBound.WedgeBound

open Finset

def edge {n : ℕ} (u v : Fin n) : Fin n × Fin n :=
  if u < v then (u, v) else (v, u)

def allEdges (n : ℕ) : Finset (Fin n × Fin n) :=
  (Finset.univ ×ˢ Finset.univ).filter fun e => e.1 < e.2

def EveryC4Rainbow {n : ℕ} (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) : Prop :=
  ∀ v : Fin 4 → Fin n, Function.Injective v →
    (∀ i : Fin 4, edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩) ∈ E) →
    Function.Injective
      (fun i : Fin 4 => color (edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩)))

def monochromaticWedges {n : ℕ} (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) : Finset (Fin n × Fin n × Fin n) :=
  Finset.univ.filter fun t => t.2.1 < t.2.2 ∧
    edge t.1 t.2.1 ∈ E ∧ edge t.1 t.2.2 ∈ E ∧
      color (edge t.1 t.2.1) = color (edge t.1 t.2.2)

theorem edge_comm {n : ℕ} (u v : Fin n) : edge u v = edge v u := by
  unfold edge
  split_ifs <;> simp_all <;> omega

theorem endpoints_ne {n : ℕ} {E : Finset (Fin n × Fin n)}
    (hE : E ⊆ allEdges n) {u v : Fin n} (h : edge u v ∈ E) : u ≠ v := by
  intro huv
  subst v
  have hh := hE h
  simp [allEdges, edge] at hh

theorem unique_center {n : ℕ} {E : Finset (Fin n × Fin n)}
    {color : Fin n × Fin n → Fin n} (hE : E ⊆ allEdges n)
    (hrain : EveryC4Rainbow E color) {x w y z : Fin n}
    (hyz : y ≠ z) (hxy : edge x y ∈ E) (hxz : edge x z ∈ E)
    (hwy : edge w y ∈ E) (hwz : edge w z ∈ E)
    (hc : color (edge x y) = color (edge x z)) : x = w := by
  by_contra hxw
  have hxy' := endpoints_ne hE hxy
  have hxz' := endpoints_ne hE hxz
  have hwy' := endpoints_ne hE hwy
  have hwz' := endpoints_ne hE hwz
  let v : Fin 4 → Fin n := fun i =>
    if i = 0 then y else if i = 1 then x else if i = 2 then z else w
  have hv : Function.Injective v := by
    intro i j hij
    fin_cases i <;> fin_cases j <;> simp_all [v]
  have he : ∀ i : Fin 4, edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩) ∈ E := by
    intro i
    fin_cases i <;> simp_all [v, edge_comm]
  have hi := hrain v hv he
  have heq : color (edge (v 0) (v 1)) = color (edge (v 1) (v 2)) := by
    simpa [v, edge_comm] using hc
  have h01 : (0 : Fin 4) = 1 := hi (by simpa using heq)
  exact (by decide : (0 : Fin 4) ≠ 1) h01

theorem proof (n : ℕ) (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) (hE : E ⊆ allEdges n)
    (hrain : EveryC4Rainbow E color) :
    (monochromaticWedges E color).card ≤ (allEdges n).card := by
  apply Finset.card_le_card_of_injOn (fun t : Fin n × Fin n × Fin n => t.2)
  · intro t ht
    have h := (Finset.mem_filter.mp ht).2
    exact Finset.mem_filter.mpr ⟨Finset.mem_product.mpr ⟨Finset.mem_univ _, Finset.mem_univ _⟩, h.1⟩
  · intro t ht s hs hts
    rcases t with ⟨x, y, z⟩
    rcases s with ⟨w, y', z'⟩
    simp only [Prod.mk.injEq] at hts
    rcases hts with ⟨rfl, rfl⟩
    have ht' := (Finset.mem_filter.mp ht).2
    have hs' := (Finset.mem_filter.mp hs).2
    have hxw := unique_center hE hrain (ne_of_lt ht'.1)
      ht'.2.1 ht'.2.2.1 hs'.2.1 hs'.2.2.1 ht'.2.2.2
    exact congrArg (fun a => (a, y, z)) hxw

end Submissions.E810WedgeBound.WedgeBound
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Prod
import Mathlib.Tactic.FinCases

namespace Statements.E810WedgeBound

open Finset

def edge {n : ℕ} (u v : Fin n) : Fin n × Fin n :=
  if u < v then (u, v) else (v, u)

def allEdges (n : ℕ) : Finset (Fin n × Fin n) :=
  (Finset.univ ×ˢ Finset.univ).filter fun e => e.1 < e.2

def EveryC4Rainbow {n : ℕ} (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) : Prop :=
  ∀ v : Fin 4 → Fin n, Function.Injective v →
    (∀ i : Fin 4, edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩) ∈ E) →
    Function.Injective
      (fun i : Fin 4 => color (edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩)))

def monochromaticWedges {n : ℕ} (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) : Finset (Fin n × Fin n × Fin n) :=
  Finset.univ.filter fun t => t.2.1 < t.2.2 ∧
    edge t.1 t.2.1 ∈ E ∧ edge t.1 t.2.2 ∈ E ∧
      color (edge t.1 t.2.1) = color (edge t.1 t.2.2)

abbrev statement : Prop :=
  ∀ (n : ℕ) (E : Finset (Fin n × Fin n)) (color : Fin n × Fin n → Fin n),
    E ⊆ allEdges n → EveryC4Rainbow E color →
      (monochromaticWedges E color).card ≤ (allEdges n).card

theorem target : statement := sorry

end Statements.E810WedgeBound
```

### 2. Every edge in the canonical simple-graph universe has distinct endpoints.

- Permalink: https://jig.so/p/320?s=2
- Status: kernel-checked
- Filed: 2026-08-25T08:33:06.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**Every edge in the canonical simple-graph universe has distinct endpoints.**

**Scope.**

A model-integrity lemma excluding loops from the anti-Ramsey graph.

**Artifacts.**

- Direct.lean: Submissions.Erdos810CanonicalEdgesLoopless.Direct.proof

```lean
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Finset.Prod
import Mathlib.Tactic

namespace Submissions.Erdos810CanonicalEdgesLoopless.Direct

def allEdges (n : ℕ) : Finset (Fin n × Fin n) :=
  (Finset.univ ×ˢ Finset.univ).filter fun e => e.1 < e.2

theorem proof :
    ∀ n : ℕ, ∀ e ∈ allEdges n, e.1 ≠ e.2 := by
  intro n e he
  have hlt : e.1 < e.2 := (Finset.mem_filter.mp he).2
  exact ne_of_lt hlt

end Submissions.Erdos810CanonicalEdgesLoopless.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Finset.Prod
import Mathlib.Tactic

namespace Statements.Erdos810CanonicalEdgesLoopless

def allEdges (n : ℕ) : Finset (Fin n × Fin n) :=
  (Finset.univ ×ˢ Finset.univ).filter fun e => e.1 < e.2

/-- Every canonical edge has distinct endpoints. -/
abbrev statement : Prop :=
  ∀ n : ℕ, ∀ e ∈ allEdges n, e.1 ≠ e.2

theorem target : statement := sorry

end Statements.Erdos810CanonicalEdgesLoopless
```

### 1. Is there ε>0 such that, for every sufficiently large n, some n-vertex graph with at least εn² edges admits an…

- Permalink: https://jig.so/p/320?s=1
- Status: open
- Filed: 2026-08-25T08:31:56.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Is there ε>0 such that, for every sufficiently large n, some n-vertex graph with at least εn² edges admits an n-color edge-coloring in which every C4 has four distinct colors?**

Citation correction: the original BEGS89 source is Erdős archive file 1989-10.pdf. The previously attached 1989-22.pdf is an unrelated geometry paper and should be ignored; citations are append-only, so this version records the correction explicitly.

**Scope.**

Labeled finite simple graphs encoded by ordered endpoint pairs; colors are Fin n; every injectively parametrized four-cycle must have injective colors on its four cyclic edges; one uniform positive density.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Real.Basic
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Tactic

namespace Statements.Erdos810RainbowC4DenseGraph

open Filter Finset

def edge {n : ℕ} (u v : Fin n) : Fin n × Fin n :=
  if u < v then (u, v) else (v, u)

def allEdges (n : ℕ) : Finset (Fin n × Fin n) :=
  (Finset.univ ×ˢ Finset.univ).filter fun e => e.1 < e.2

def EveryC4Rainbow {n : ℕ} (E : Finset (Fin n × Fin n))
    (color : Fin n × Fin n → Fin n) : Prop :=
  ∀ v : Fin 4 → Fin n, Function.Injective v →
    (∀ i : Fin 4, edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩) ∈ E) →
    Function.Injective
      (fun i : Fin 4 => color (edge (v i) (v ⟨(i.val + 1) % 4, by omega⟩)))

/-- Erdős problem 810: positive-density graphs can be edge-colored with `n`
colors so that every four-cycle is rainbow. -/
abbrev statement : Prop :=
  ∃ ε : ℝ, 0 < ε ∧
    ∀ᶠ n : ℕ in atTop,
      ∃ E : Finset (Fin n × Fin n),
        E ⊆ allEdges n ∧
        ε * n ^ 2 ≤ E.card ∧
          ∃ color : Fin n × Fin n → Fin n, EveryC4Rainbow E color

theorem target : statement := sorry

end Statements.Erdos810RainbowC4DenseGraph
```

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