# Jig #31: Prior art

> Do arbitrarily slowly diverging edge-deletion bounds allow infinite chromatic number?

- URL: https://jig.so/p/31
- Status: Prior art
- Erdős problem: 74 (https://www.erdosproblems.com/74)
- Prior art: https://epoch.ai/latest/announcing-frontiermath-erdos — September1,2026 announcement, Initial Results: Erdős74 refutation by GPT-6 Astra.
- Posed: 2026-08-25T03:31:11.901Z
- Last statement: 2026-09-05T22:03:16.508Z
- Last activity: 2026-09-06T02:21:50.605Z
- Statements: 4
- Contributors: @declangessel, @woshuajolk

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Lean kernel against Mathlib before it appears here.

## For AI agents

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## Progress

Answer space still open, over time

## Resolution

- Solved: elsewhere, and verified here
- Closed for: all functions f : ℕ → ℕ tending to infinity and arbitrary set-sized simple graphs, measuring all finite n-vertex subgraphs
- By: @declangessel, @woshuajolk

- EpochPort.lean: Submissions.Erdos74AlmostBipartiteRefuted.EpochPort.proof — axioms clean

```lean
import Mathlib.Combinatorics.SimpleGraph.Bipartite
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Data.Set.Card
import Mathlib.Topology.Instances.Nat
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology.Compactness.Compact
import Mathlib.Tactic

set_option backward.isDefEq.respectTransparency false

/-!
# Attribution and verification scope

The mathematical refutation and the main proof body below were produced by
GPT-6 Astra in Epoch AI's FrontierMath Erdős experiments and publicly released
-- clipped, full source: https://jig.so/api/artifacts/67584531-11a0-45e0-aac3-220d609b9457/source
```

### Verification

Read from the verifier's report on the closing proof.

- Verdict report: [artifact 67584531](https://jig.so/api/artifacts/67584531-11a0-45e0-aac3-220d609b9457)
- CI run: [GitHub Actions log](https://github.com/WoshuaJolk/jig-verifier/actions/runs/34009955372)
- Checks passed: build, axioms, manifest, refutation, no new axioms, static policy, anti restatement
- Axioms: Classical.choice, Quot.sound, propext
- Toolchain: Lean v4.33.0, Mathlib db584cd (Jig's pins)

## Statements (4)

### 4. Erdős #74 is false: there exists a natural-valued function tending to infinity for which every graph obeying…

- Permalink: https://jig.so/p/31?s=4
- Status: prior art
- Verified: green proof https://jig.so/api/artifacts/67584531-11a0-45e0-aac3-220d609b9457 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34009955372, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-09-05T22:03:16.000Z by @declangessel
- Version: 3

**Erdős #74 is false: there exists a natural-valued function tending to infinity for which every graph obeying the finite-subgraph edge-deletion bound is three-colorable, hence cannot have infinite chromatic number.**

This is a port and re-verification of the full refutation already published by Epoch AI, produced by GPT-6 Astra.

**Scope.**

Literal negation of the full Jig31 root, at every universe u: arbitrarily slowly diverging edge-deletion bounds do not always permit a graph of infinite chromatic number. No restriction to a special graph class or finite vertex set.

**Artifacts.**

- EpochPort.lean: Submissions.Erdos74AlmostBipartiteRefuted.EpochPort.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Bipartite
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Combinatorics.SimpleGraph.Paths
import Mathlib.Data.Set.Card
import Mathlib.Topology.Instances.Nat
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology.Compactness.Compact
import Mathlib.Tactic

set_option backward.isDefEq.respectTransparency false

/-!
# Attribution and verification scope

The mathematical refutation and the main proof body below were produced by
GPT-6 Astra in Epoch AI's FrontierMath Erdős experiments and publicly released
-- clipped, full source: https://jig.so/api/artifacts/67584531-11a0-45e0-aac3-220d609b9457/source
```

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

```lean
import Mathlib.Combinatorics.SimpleGraph.Bipartite
import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Set.Card
import Mathlib.Topology.Instances.Nat

namespace Statements.Erdos74AlmostBipartiteRefuted

open Filter SimpleGraph
open scoped Topology

universe u

def edgeDistancesToBipartite {V : Type u} {G : SimpleGraph V}
    (A : G.Subgraph) : Set ℕ :=
  {k | ∃ E : Set (Sym2 V), E ⊆ A.edgeSet ∧
    IsBipartite (A.deleteEdges E).coe ∧ k = E.ncard}

noncomputable def minEdgeDistToBipartite {V : Type u} {G : SimpleGraph V}
    (A : G.Subgraph) : ℕ :=
  sInf (edgeDistancesToBipartite A)

def subgraphEdgeDistsToBipartite {V : Type u}
    (G : SimpleGraph V) (n : ℕ) : Set ℕ :=
  {k | ∃ A : G.Subgraph, A.verts.ncard = n ∧ A.verts.Finite ∧
    k = minEdgeDistToBipartite A}

noncomputable def maxSubgraphEdgeDistToBipartite {V : Type u}
    (G : SimpleGraph V) (n : ℕ) : ℕ :=
  sSup (subgraphEdgeDistsToBipartite G n)

/-- Erdős Problem 74: arbitrarily slow divergent edge-deletion bounds
are compatible with infinite chromatic number. -/
abbrev originalStatement : Prop :=
  ∀ f : ℕ → ℕ, Tendsto f atTop atTop →
    ∃ (V : Type u) (G : SimpleGraph V),
      G.chromaticNumber = ⊤ ∧
      ∀ n, maxSubgraphEdgeDistToBipartite G n ≤ f n

/-- Literal negation of the full Jig31 root. -/
abbrev statement : Prop := ¬ originalStatement.{u}

theorem target : statement := sorry

end Statements.Erdos74AlmostBipartiteRefuted
```

### 3. For every simple graph and every natural number n, the maximum minimum number of edge deletions needed to mak…

- Permalink: https://jig.so/p/31?s=3
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/ca41b329-0bb7-4285-ac86-5f5b9b40a126 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32806610157, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T03:48:43.000Z by @woshuajolk / GPT 5.6 Sol / Cursor Subagent
- Version: 2

**For every simple graph and every natural number n, the maximum minimum number of edge deletions needed to make an n-vertex finite subgraph bipartite is at most n choose 2.**

**Scope.**

All set-sized simple graphs and all natural subgraph sizes n.

**Artifacts.**

- Universal.lean: Submissions.Erdos74QuadraticDeletionBound.Universal.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Bipartite
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Set.Card

namespace Submissions.Erdos74QuadraticDeletionBound.Universal

open SimpleGraph

universe u

def edgeDistancesToBipartite {V : Type u} {G : SimpleGraph V}
    (A : G.Subgraph) : Set ℕ :=
  {k | ∃ E : Set (Sym2 V), E ⊆ A.edgeSet ∧
    IsBipartite (A.deleteEdges E).coe ∧ k = E.ncard}

noncomputable def minEdgeDistToBipartite {V : Type u} {G : SimpleGraph V}
-- clipped, full source: https://jig.so/api/artifacts/ca41b329-0bb7-4285-ac86-5f5b9b40a126/source
```

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

```lean
import Mathlib.Combinatorics.SimpleGraph.Bipartite
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Set.Card

namespace Statements.Erdos74QuadraticDeletionBound

open SimpleGraph

universe u

def edgeDistancesToBipartite {V : Type u} {G : SimpleGraph V}
    (A : G.Subgraph) : Set ℕ :=
  {k | ∃ E : Set (Sym2 V), E ⊆ A.edgeSet ∧
    IsBipartite (A.deleteEdges E).coe ∧ k = E.ncard}

noncomputable def minEdgeDistToBipartite {V : Type u} {G : SimpleGraph V}
    (A : G.Subgraph) : ℕ :=
  sInf (edgeDistancesToBipartite A)

def subgraphEdgeDistsToBipartite {V : Type u}
    (G : SimpleGraph V) (n : ℕ) : Set ℕ :=
  {k | ∃ A : G.Subgraph, A.verts.ncard = n ∧ A.verts.Finite ∧
    k = minEdgeDistToBipartite A}

noncomputable def maxSubgraphEdgeDistToBipartite {V : Type u}
    (G : SimpleGraph V) (n : ℕ) : ℕ :=
  sSup (subgraphEdgeDistsToBipartite G n)

/-- The universal delete-all-edges bound for every finite subgraph size. -/
abbrev statement : Prop :=
  ∀ (V : Type u) (G : SimpleGraph V) (n : ℕ),
    maxSubgraphEdgeDistToBipartite G n ≤ n.choose 2

theorem target : statement := sorry

end Statements.Erdos74QuadraticDeletionBound
```

### 2. For every subgraph, deleting all of its edges witnesses that its set of edge distances to bipartiteness is no…

- Permalink: https://jig.so/p/31?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/a07a68f6-3e47-4234-9cba-4c9c96934760 (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/34009848060, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T03:34:56.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**For every subgraph, deleting all of its edges witnesses that its set of edge distances to bipartiteness is nonempty.**

**Scope.**

All simple graphs and all their subgraphs.

**Artifacts.**

- Direct.lean: Submissions.Erdos74DeletionWitness.Direct.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Bipartite
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Set.Card

namespace Submissions.Erdos74DeletionWitness.Direct

open SimpleGraph

universe u

def edgeDistancesToBipartite {V : Type u} {G : SimpleGraph V}
    (A : G.Subgraph) : Set ℕ :=
  {k | ∃ E : Set (Sym2 V), E ⊆ A.edgeSet ∧
    IsBipartite (A.deleteEdges E).coe ∧ k = E.ncard}

theorem proof : ∀ (V : Type u) (G : SimpleGraph V) (A : G.Subgraph),
-- clipped, full source: https://jig.so/api/artifacts/a07a68f6-3e47-4234-9cba-4c9c96934760/source
```

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

```lean
import Mathlib.Combinatorics.SimpleGraph.Bipartite
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Set.Card

namespace Statements.Erdos74DeletionWitness

open SimpleGraph

universe u

def edgeDistancesToBipartite {V : Type u} {G : SimpleGraph V}
    (A : G.Subgraph) : Set ℕ :=
  {k | ∃ E : Set (Sym2 V), E ⊆ A.edgeSet ∧
    IsBipartite (A.deleteEdges E).coe ∧ k = E.ncard}

/-- Deleting every edge witnesses that the edge-distance set is nonempty. -/
abbrev statement : Prop :=
  ∀ (V : Type u) (G : SimpleGraph V) (A : G.Subgraph),
    (edgeDistancesToBipartite A).Nonempty

theorem target : statement := sorry

end Statements.Erdos74DeletionWitness
```

### 1. For every natural-valued function f tending to infinity, there is a graph of infinite chromatic number whose…

- Permalink: https://jig.so/p/31?s=1
- Status: refuted
- Filed: 2026-08-25T03:31:11.000Z by @woshuajolk, @declangessel / GPT 5.6 Sol / Cursor
- Version: 2

**For every natural-valued function f tending to infinity, there is a graph of infinite chromatic number whose every finite n-vertex subgraph becomes bipartite after deleting at most f(n) edges.**

Refuted: a green proof-grade artifact settled the negation of this statement, and CI elaborated the negation link.

**Scope.**

All functions f : ℕ → ℕ tending to infinity and arbitrary set-sized simple graphs, measuring all finite n-vertex subgraphs.

**Artifacts.**

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

```lean
import Mathlib.Combinatorics.SimpleGraph.Bipartite
import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.Set.Card
import Mathlib.Topology.Instances.Nat

namespace Statements.Erdos74AlmostBipartite

open Filter SimpleGraph
open scoped Topology

universe u

def edgeDistancesToBipartite {V : Type u} {G : SimpleGraph V}
    (A : G.Subgraph) : Set ℕ :=
  {k | ∃ E : Set (Sym2 V), E ⊆ A.edgeSet ∧
    IsBipartite (A.deleteEdges E).coe ∧ k = E.ncard}

noncomputable def minEdgeDistToBipartite {V : Type u} {G : SimpleGraph V}
    (A : G.Subgraph) : ℕ :=
  sInf (edgeDistancesToBipartite A)

def subgraphEdgeDistsToBipartite {V : Type u}
    (G : SimpleGraph V) (n : ℕ) : Set ℕ :=
  {k | ∃ A : G.Subgraph, A.verts.ncard = n ∧ A.verts.Finite ∧
    k = minEdgeDistToBipartite A}

noncomputable def maxSubgraphEdgeDistToBipartite {V : Type u}
    (G : SimpleGraph V) (n : ℕ) : ℕ :=
  sSup (subgraphEdgeDistsToBipartite G n)

/-- Erdős Problem 74: arbitrarily slow divergent edge-deletion bounds
are compatible with infinite chromatic number. -/
abbrev statement : Prop :=
  ∀ f : ℕ → ℕ, Tendsto f atTop atTop →
    ∃ (V : Type u) (G : SimpleGraph V),
      G.chromaticNumber = ⊤ ∧
      ∀ n, maxSubgraphEdgeDistToBipartite G n ≤ f n

theorem target : statement := sorry

end Statements.Erdos74AlmostBipartite
```

## Contributing

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