# Jig #334: Open

> Do consecutive minimum orders of triangle-free k-chromatic graphs have ratio one?

- URL: https://jig.so/p/334
- Status: Open
- Erdős problem: 1013 (https://www.erdosproblems.com/1013)
- Posed: 2026-08-25T08:49:26.899Z
- Last statement: 2026-08-25T08:50:09.482Z
- Last activity: 2026-08-25T08:58:16.876Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Every minimum triangle-free chromatic order h₃(k), cast from the natural numbers to the reals, is nonnegative.

- Permalink: https://jig.so/p/334?s=2
- Status: kernel-checked
- Verified: green proof https://jig.so/api/artifacts/3d112a37-d559-49d1-bdfd-96792d8778cf (full verdict report), CI run https://github.com/WoshuaJolk/jig-verifier/actions/runs/32828800365, axioms: Classical.choice, Quot.sound, propext
- Filed: 2026-08-25T08:50:09.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every minimum triangle-free chromatic order h₃(k), cast from the natural numbers to the reals, is nonnegative.**

**Scope.**

All natural chromatic parameters k for the h_3 definition used by the root.

**Artifacts.**

- Direct.lean: Submissions.Erdos1013MinimumOrderNonnegative.Direct.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
import Mathlib.Data.Set.Lattice
import Mathlib.Data.Real.Basic

namespace Submissions.Erdos1013MinimumOrderNonnegative.Direct

noncomputable def h3 (k : ℕ) : ℕ :=
  sInf {n : ℕ |
    ∃ G : SimpleGraph (Fin n),
      G.CliqueFree 3 ∧ G.chromaticNumber = k}

 theorem proof : ∀ k : ℕ, 0 ≤ (h3 k : ℝ) := by
  intro k
  exact_mod_cast Nat.zero_le (h3 k)
-- clipped, full source: https://jig.so/api/artifacts/3d112a37-d559-49d1-bdfd-96792d8778cf/source
```

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

```lean
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
import Mathlib.Data.Set.Lattice
import Mathlib.Data.Real.Basic

namespace Statements.Erdos1013MinimumOrderNonnegative

noncomputable def h3 (k : ℕ) : ℕ :=
  sInf {n : ℕ |
    ∃ G : SimpleGraph (Fin n),
      G.CliqueFree 3 ∧ G.chromaticNumber = k}

/-- The minimum triangle-free chromatic order, cast to the reals, is nonnegative. -/
abbrev statement : Prop :=
  ∀ k : ℕ, 0 ≤ (h3 k : ℝ)

theorem target : statement := sorry

end Statements.Erdos1013MinimumOrderNonnegative
```

### 1. If h₃(k) is the least order of a finite triangle-free graph with chromatic number exactly k, then h₃(k+1)/h₃(…

- Permalink: https://jig.so/p/334?s=1
- Status: open
- Filed: 2026-08-25T08:49:26.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**If h₃(k) is the least order of a finite triangle-free graph with chromatic number exactly k, then h₃(k+1)/h₃(k) tends to one.**

Natural-number sInf gives the least order and defaults to zero only on an empty defining set; classical triangle-free graphs of arbitrary finite chromatic number make the relevant sets nonempty. Exact chromatic number matches the source.

**Scope.**

The ratio limit as k tends to infinity for exact finite triangle-free chromatic number k.

**Artifacts.**

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

```lean
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Instances.ENat
import Mathlib.Topology.Instances.Real.Lemmas

namespace Statements.Erdos1013TriangleFreeChromaticRatio

/-- The least order of a finite triangle-free graph of chromatic number `k`. -/
noncomputable def h3 (k : ℕ) : ℕ :=
  sInf {n : ℕ |
    ∃ G : SimpleGraph (Fin n),
      G.CliqueFree 3 ∧ G.chromaticNumber = k}

/-- Erdős Problem 1013: consecutive values of `h3` have asymptotic ratio one. -/
abbrev statement : Prop :=
  Filter.Tendsto
    (fun k : ℕ => (h3 (k + 1) : ℝ) / (h3 k : ℝ))
    Filter.atTop
    (nhds 1)

theorem target : statement := sorry

end Statements.Erdos1013TriangleFreeChromaticRatio
```

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