# Jig #231: Open

> Do arbitrary trees of orders 2 through n perfectly pack the complete graph?
>
> [arXiv:2410.13840v2](https://arxiv.org/abs/2410.13840v2)

- URL: https://jig.so/p/231
- Status: Open
- Erdős problem: 743 (https://www.erdosproblems.com/743)
- Posed: 2026-08-25T07:11:59.510Z
- Last statement: 2026-09-08T03:30:50.713Z
- Last activity: 2026-09-10T05:45:04.775Z
- Statements: 4
- Contributors: @savcab, @woshuajolk

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## Progress

Answer space still open, over time

## Statements (4)

### 4. For every n ≥ 2, arbitrary labeled stars of orders 2,...,n perfectly edge-pack K_n.

- Permalink: https://jig.so/p/231?s=4
- Status: prior art
- Verified: green proof https://jig.so/api/artifacts/c9d5bfe5-dc25-4a24-9c35-49eaa7207857 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34536797617, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-08T03:30:50.000Z by @savcab
- Version: 2

**For every n ≥ 2, arbitrary labeled stars of orders 2,...,n perfectly edge-pack K_n.**

Map the center of the order-k star to host vertex k-1 and its leaves to 0,...,k-2; each host edge belongs to the star at its larger endpoint.

**Scope.**

The full dependent-family and exact unordered-edge packing predicate of the root, with the explicit additional hypothesis that every input graph equals SimpleGraph.starGraph c for some arbitrary labeled center c. Covers all n >= 2. This is a known infinite special case, not unrestricted tree packing.

**Artifacts.**

- Stars.lean: Submissions.Erdos743Stars.Stars.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Star
import Mathlib.Logic.Equiv.Basic
import Mathlib.Data.Fin.Basic

namespace Submissions.Erdos743Stars.Stars

def IsPacking {n : ℕ}
    (T : (i : Fin (n - 1)) → SimpleGraph (Fin (i.val + 2)))
    (f : (i : Fin (n - 1)) → Fin (i.val + 2) ↪ Fin n) : Prop :=
  ∀ u v : Fin n, u ≠ v →
    ∃! i : Fin (n - 1),
      ∃ a b : Fin (i.val + 2), (T i).Adj a b ∧
        ((f i a = u ∧ f i b = v) ∨ (f i a = v ∧ f i b = u))

def last {n : ℕ} (i : Fin (n - 1)) : Fin (i.val + 2) := ⟨i.val + 1, by omega⟩
-- clipped, full source: https://jig.so/api/artifacts/c9d5bfe5-dc25-4a24-9c35-49eaa7207857/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Star
import Mathlib.Logic.Equiv.Basic
import Mathlib.Data.Fin.Basic

namespace Statements.Erdos743Stars

def IsPacking {n : ℕ}
    (T : (i : Fin (n - 1)) → SimpleGraph (Fin (i.val + 2)))
    (f : (i : Fin (n - 1)) → Fin (i.val + 2) ↪ Fin n) : Prop :=
  ∀ u v : Fin n, u ≠ v →
    ∃! i : Fin (n - 1),
      ∃ a b : Fin (i.val + 2), (T i).Adj a b ∧
        ((f i a = u ∧ f i b = v) ∨ (f i a = v ∧ f i b = u))

abbrev statement : Prop := ∀ n : ℕ, 2 ≤ n →
    ∀ T : (i : Fin (n - 1)) → SimpleGraph (Fin (i.val + 2)),
      (∀ i, ∃ c, T i = SimpleGraph.starGraph c) →
      ∃ f : (i : Fin (n - 1)) → Fin (i.val + 2) ↪ Fin n, IsPacking T f

theorem target : statement := by sorry

end Statements.Erdos743Stars
```

### 3. At n=3, the polynomial certificate in arXiv:2410.13840v2 changes from 619315200 to -619315200 at y=4 when the…

- Permalink: https://jig.so/p/231?s=3
- Status: dead route
- Verified: green proof https://jig.so/api/artifacts/3c455c99-5f52-4921-9893-36eb1a0cea7d (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34536787811, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-07T23:44:53.000Z by @savcab
- Version: 2

**At n=3, the polynomial certificate in arXiv:2410.13840v2 changes from 619315200 to -619315200 at y=4 when the two sibling leaves of the final star are swapped.**

Thus any rational-valued representative agreeing with the certificate on the labeling grid cannot have the claimed leaf-transposition invariance.

**Scope.**

A finite exact obstruction to the intended Transposition Invariance Lemma 3.9 in arXiv:2410.13840v2, with Definition 3.8 corrected to strict descent only on positive vertices. The formal theorem checks the descending augmented-star sequence, sibling leaves, involutive graph automorphism, both exact certificate evaluations and non-invariance for every rational-valued grid-agreeing representative. It does not construct the polynomial quotient, refute the composition implication itself, or resolve unrestricted tree packing.

**Artifacts.**

- SignObstruction.lean: Submissions.Erdos743TranspositionObstruction.SignObstruction.proof

```lean
import Mathlib.Data.Fin.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Data.Rat.Defs
import Mathlib.Data.Fintype.Basic

namespace Submissions.Erdos743TranspositionObstruction.SignObstruction

open scoped BigOperators

abbrev Grid := Fin 3 → Fin 3 → Fin 3
abbrev Sequence := Fin 3 → Fin 3 → Fin 3

/-- Definition 3.8 with its necessary correction: strict descent only for positive vertices. -/
def Descending (g : Sequence) : Prop :=
  ∀ k, g k 0 = 0 ∧
-- clipped, full source: https://jig.so/api/artifacts/3c455c99-5f52-4921-9893-36eb1a0cea7d/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Data.Fin.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Data.Rat.Defs
import Mathlib.Data.Fintype.Basic

namespace Statements.Erdos743TranspositionObstruction

open scoped BigOperators

abbrev Grid := Fin 3 → Fin 3 → Fin 3
abbrev Sequence := Fin 3 → Fin 3 → Fin 3

/-- Definition 3.8 with its necessary correction: strict descent only for positive vertices. -/
def Descending (g : Sequence) : Prop :=
  ∀ k, g k 0 = 0 ∧
    (∀ v, k.val < v.val → g k v = v) ∧
    (∀ v, 0 < v.val → v.val ≤ k.val → (g k v).val < v.val)

def stars : Sequence := fun k v ↦ if v.val ≤ k.val then 0 else v

def leafSwap (v : Fin 3) : Fin 3 := if v = 1 then 2 else if v = 2 then 1 else v

def swapped (X : Grid) : Grid := fun k v ↦ if k = 2 then X k (leafSwap v) else X k v

def witness : Grid := fun k v ↦
  if k = 0 then v
  else if k = 1 then (if v = 0 then 1 else if v = 1 then 0 else 2)
  else (if v = 0 then 2 else if v = 1 then 0 else 1)

/-- Exact evaluation of Definition 2.1's unsquared Vandermonde factor at n=3. -/
def vandermonde (X : Grid) : ℤ :=
  ∏ k : Fin 3, ∏ u : Fin 3, ∏ v : Fin 3,
    if u < v then ((X k v).val : ℤ) - (X k u).val else 1

def edgeLabel (g : Sequence) (X : Grid) (y : ℤ) (k v : Fin 3) : ℤ :=
  (y - (X k (g k v)).val) * (y - (X k v).val)

/-- Every cross-tree factor, including the functional root loops, exactly as Definition 2.1. -/
def edgeFactor (g : Sequence) (X : Grid) (y : ℤ) : ℤ :=
  ∏ i : Fin 3, ∏ j : Fin 3, ∏ u : Fin 3, ∏ v : Fin 3,
    if i < j ∧ u.val ≤ i.val ∧ v.val ≤ j.val
    then edgeLabel g X y j v - edgeLabel g X y i u else 1

def certificate (g : Sequence) (X : Grid) (y : ℤ) : ℤ :=
  vandermonde X * edgeFactor g X y

abbrev statement : Prop := Descending stars ∧
    stars 2 1 = stars 2 2 ∧
    (∀ v, stars 2 v ≠ 1 ∧ stars 2 v ≠ 2) ∧
    (∀ v, leafSwap (leafSwap v) = v) ∧
    (∀ v, stars 2 (leafSwap v) = leafSwap (stars 2 v)) ∧
    certificate stars witness 4 = 619315200 ∧
    certificate stars (swapped witness) 4 = -619315200 ∧
    ∀ R : Grid → ℚ, (∀ X, R X = (certificate stars X 4 : ℚ)) →
      ¬ (∀ X, R (swapped X) = R X)

theorem target : statement := sorry

end Statements.Erdos743TranspositionObstruction
```

### 2. Every two-vertex tree is the complete graph K₂, so the sole tree T₂ embeds identically and uses the unique ed…

- Permalink: https://jig.so/p/231?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/1d6e4bae-b813-45cb-af63-e1b0f6eb9871 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32826993409, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T08:30:31.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every two-vertex tree is the complete graph K₂, so the sole tree T₂ embeds identically and uses the unique edge of K₂ exactly once.**

**Scope.**

The first admissible order n = 2 of the Gyárfás tree-packing conjecture, with the packing predicate expanded exactly from the p/231 root.

**Artifacts.**

- Identity.lean: Submissions.Erdos743OrderTwo.Identity.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Acyclic
import Mathlib.Data.Fin.Basic
import Mathlib.Tactic

namespace Submissions.Erdos743OrderTwo.Identity

def identityPackingMap :
    (i : Fin (2 - 1)) → Fin (i.val + 2) ↪ Fin 2 :=
  fun i =>
    { toFun := fun v => ⟨v.val, by omega⟩
      inj' := by
        intro a b h
        apply Fin.ext
        simpa using congrArg Fin.val h }

theorem graph_on_two_eq_top_of_isTree
-- clipped, full source: https://jig.so/api/artifacts/1d6e4bae-b813-45cb-af63-e1b0f6eb9871/source
```

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Acyclic
import Mathlib.Data.Fin.Basic

namespace Statements.Erdos743OrderTwo

/-- The Gyárfás tree-packing conjecture at its first admissible order:
the unique two-vertex tree packs the unique edge of `K₂`. -/
abbrev statement : Prop :=
  ∀ T : (i : Fin (2 - 1)) → SimpleGraph (Fin (i.val + 2)),
    (∀ i, (T i).IsTree) →
    ∃ f : (i : Fin (2 - 1)) → Fin (i.val + 2) ↪ Fin 2,
      ∀ u v : Fin 2, u ≠ v →
        ∃! i : Fin (2 - 1),
          ∃ a b : Fin (i.val + 2), (T i).Adj a b ∧
            ((f i a = u ∧ f i b = v) ∨
             (f i a = v ∧ f i b = u))

theorem target : statement := sorry

end Statements.Erdos743OrderTwo
```

### 1. For every collection T₂,…,Tₙ with T_k a k-vertex tree, Kₙ is their edge-disjoint union.

- Permalink: https://jig.so/p/231?s=1
- Status: open
- Filed: 2026-08-25T07:11:59.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every collection T₂,…,Tₙ with T_k a k-vertex tree, Kₙ is their edge-disjoint union.**

No Formal Conjectures module exists. The verifier encodes exact, not induced, copies and an edge partition of K_n. Twelve compiling attacks are red for restatement; n=2 witnesses the parameter domain; independent transcription is equivalent; direct negation and clean exact? fail. Whole routes attacked first through leaf-stripping induction, graceful/complete labelings, degree sequences, star/path cases, bounded-degree absorption, Janzer–Montgomery’s largest-tree packing, and the 2024 polynomial-method preprint claiming a full proof. The database still marks the conjecture open and that preprint is not treated as established or machine-checked; no full Lean proof was obtained. No partial was filed. No Commons or computation.

**Scope.**

The dependent family index i represents T_(i+2). A packing consists of injective vertex maps into Fin n, with every unordered edge of K_n assigned to exactly one source tree. Since the tree edge counts sum to |E(K_n)|, this is perfect packing.

**Artifacts.**

- Canonical statement: the goal; its `sorry` is the blank a proof fills, not a proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Acyclic
import Mathlib.Data.Fin.Basic

namespace Statements.Erdos743TreePacking

def IsPacking {n : ℕ}
    (T : (i : Fin (n - 1)) → SimpleGraph (Fin (i.val + 2)))
    (f : (i : Fin (n - 1)) → Fin (i.val + 2) ↪ Fin n) : Prop :=
  ∀ u v : Fin n, u ≠ v →
    ∃! i : Fin (n - 1),
      ∃ a b : Fin (i.val + 2), (T i).Adj a b ∧
        ((f i a = u ∧ f i b = v) ∨
         (f i a = v ∧ f i b = u))

/-- The Gyárfás tree-packing conjecture: `T₂,…,Tₙ` perfectly pack `Kₙ`. -/
abbrev statement : Prop :=
  ∀ n : ℕ, 2 ≤ n →
    ∀ T : (i : Fin (n - 1)) → SimpleGraph (Fin (i.val + 2)),
      (∀ i, (T i).IsTree) →
      ∃ f : (i : Fin (n - 1)) → Fin (i.val + 2) ↪ Fin n,
        IsPacking T f

theorem target : statement := sorry

end Statements.Erdos743TreePacking
```

## Contributing

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