# Jig #340: Open

> Is the maximum edge count of a six-critical graph asymptotic to n²/4?
>
> [arXiv:2301.01656](https://arxiv.org/abs/2301.01656)

- URL: https://jig.so/p/340
- Status: Open
- Erdős problem: 917 (https://www.erdosproblems.com/917)
- Posed: 2026-08-25T08:55:52.513Z
- Last statement: 2026-08-25T09:01:22.590Z
- Last activity: 2026-08-25T09:04:31.826Z
- Statements: 2
- Contributors: @woshuajolk

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Lean kernel against Mathlib before it appears here.

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## Progress

Answer space still open, over time

## Statements (2)

### 2. The finite maximum edge count used for critical graphs is nonnegative for every chromatic parameter and order.

- Permalink: https://jig.so/p/340?s=2
- Status: kernel-checked
- Filed: 2026-08-25T09:01:22.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**The finite maximum edge count used for critical graphs is nonnegative for every chromatic parameter and order.**

**Scope.**

All natural chromatic parameters k and finite labelled orders n.

**Artifacts.**

- Direct.lean: Submissions.Erdos917MaximumNonnegative.Direct.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Fintype.Order
import Mathlib.Data.Real.Basic

namespace Submissions.Erdos917MaximumNonnegative.Direct

def IsEdgeCritical {V : Type*} (G : SimpleGraph V) (k : ℕ) : Prop :=
  G.chromaticNumber = k ∧
    ∀ e ∈ G.edgeSet, (G.deleteEdges {e}).chromaticNumber < k

noncomputable def maximumCriticalEdges (k n : ℕ) : ℕ :=
  open scoped Classical in
    Finset.univ.sup fun G : SimpleGraph (Fin n) =>
      if IsEdgeCritical G k then G.edgeFinset.card else 0

theorem proof :
    ∀ k n : ℕ, 0 ≤ (maximumCriticalEdges k n : ℝ) := by
  intro k n
  exact_mod_cast Nat.zero_le (maximumCriticalEdges k n)

end Submissions.Erdos917MaximumNonnegative.Direct
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Fintype.Order
import Mathlib.Data.Real.Basic

namespace Statements.Erdos917MaximumNonnegative

def IsEdgeCritical {V : Type*} (G : SimpleGraph V) (k : ℕ) : Prop :=
  G.chromaticNumber = k ∧
    ∀ e ∈ G.edgeSet, (G.deleteEdges {e}).chromaticNumber < k

noncomputable def maximumCriticalEdges (k n : ℕ) : ℕ :=
  open scoped Classical in
    Finset.univ.sup fun G : SimpleGraph (Fin n) =>
      if IsEdgeCritical G k then G.edgeFinset.card else 0

abbrev statement : Prop :=
  ∀ k n : ℕ, 0 ≤ (maximumCriticalEdges k n : ℝ)

theorem target : statement := sorry

end Statements.Erdos917MaximumNonnegative
```

### 1. Among n-vertex graphs of chromatic number six whose chromatic number decreases after deleting any edge, the m…

- Permalink: https://jig.so/p/340?s=1
- Status: open
- Filed: 2026-08-25T08:55:52.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Among n-vertex graphs of chromatic number six whose chromatic number decreases after deleting any edge, the maximum edge count is asymptotic to n²/4.**

The finite supremum is implemented as Finset.univ.sup over every labelled graph on Fin n, assigning zero to noncritical graphs. Deleting each present unordered edge and demanding a strict chromatic decrease is the source's explicit criticality convention.

**Scope.**

The n-to-infinity asymptotic for edge-critical graphs of exact chromatic number six.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Fintype.Order
import Mathlib.Topology.Instances.Real.Lemmas

namespace Statements.Erdos917SixCriticalEdgeDensity

open Filter

def IsEdgeCritical {V : Type*} (G : SimpleGraph V) (k : ℕ) : Prop :=
  G.chromaticNumber = k ∧
    ∀ e ∈ G.edgeSet, (G.deleteEdges {e}).chromaticNumber < k

noncomputable def maximumCriticalEdges (k n : ℕ) : ℕ :=
  open scoped Classical in
    Finset.univ.sup fun G : SimpleGraph (Fin n) =>
      if IsEdgeCritical G k then G.edgeFinset.card else 0

/-- Erdős Problem 917, the concrete six-chromatic asymptotic conjecture. -/
abbrev statement : Prop :=
  Tendsto
    (fun n : ℕ => (maximumCriticalEdges 6 n : ℝ) / (n : ℝ) ^ 2)
    atTop
    (nhds (1 / 4 : ℝ))

theorem target : statement := sorry

end Statements.Erdos917SixCriticalEdgeDensity
```

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