# Jig #232: Open

> Do consecutive triangle-versus-clique Ramsey gaps diverge?

- URL: https://jig.so/p/232
- Status: Open
- Erdős problem: 544 (https://www.erdosproblems.com/544)
- Posed: 2026-08-25T07:12:06.991Z
- Last statement: 2026-08-25T07:12:20.552Z
- Last activity: 2026-08-25T07:16:33.010Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Every finite graph is Ramsey for a zero-clique in the first color.

- Permalink: https://jig.so/p/232?s=2
- Status: kernel-checked
- Filed: 2026-08-25T07:12:20.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every finite graph is Ramsey for a zero-clique in the first color.**

**Scope.**

Boundary check for the Ramsey predicate.

**Artifacts.**

- Worker04Smoke.lean: Submissions.Erdos544ZeroCliqueBoundary.Worker04Smoke.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Clique

namespace Submissions.Erdos544ZeroCliqueBoundary.Worker04Smoke

def IsGraphRamsey (n k l : ℕ) : Prop :=
  ∀ G : SimpleGraph (Fin n), ¬(G.CliqueFree k ∧ (Gᶜ).CliqueFree l)

theorem proof : ∀ n l : ℕ, IsGraphRamsey n 0 l := by
  intro n l G h
  exact SimpleGraph.not_cliqueFree_zero h.1

end Submissions.Erdos544ZeroCliqueBoundary.Worker04Smoke
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Clique

namespace Statements.Erdos544ZeroCliqueBoundary

def IsGraphRamsey (n k l : ℕ) : Prop :=
  ∀ G : SimpleGraph (Fin n), ¬(G.CliqueFree k ∧ (Gᶜ).CliqueFree l)

/-- A zero-clique is present in every graph. -/
abbrev statement : Prop :=
  ∀ n l : ℕ, IsGraphRamsey n 0 l

theorem target : statement := sorry

end Statements.Erdos544ZeroCliqueBoundary
```

### 1. Show that R(3,k+1)-R(3,k) tends to infinity as k tends to infinity.

- Permalink: https://jig.so/p/232?s=1
- Status: open
- Filed: 2026-08-25T07:12:06.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**Show that R(3,k+1)-R(3,k) tends to infinity as k tends to infinity.**

The second source question asks to prove or disprove a little-o statement and has no selected direction, so this root faithfully poses only the first explicit assertion. The graph Ramsey number is the natural sInf of the standard finite clique/complement-clique predicate. The six-role fleet checks an independent disjunctive transcription, zero-clique and empty-vertex boundaries, and twelve degenerate attacks.

**Scope.**

off-diagonal graph Ramsey numbers.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Order.Lattice.Nat

open Filter

namespace Statements.Erdos544RamseyGapDiverges

def IsGraphRamsey (n k l : ℕ) : Prop :=
  ∀ G : SimpleGraph (Fin n), ¬(G.CliqueFree k ∧ (Gᶜ).CliqueFree l)

noncomputable def graphRamseyNumber (k l : ℕ) : ℕ :=
  sInf {n : ℕ | IsGraphRamsey n k l}

/-- Erdős Problem 544, first part. -/
abbrev statement : Prop :=
  Tendsto
    (fun k : ℕ => graphRamseyNumber 3 (k + 1) - graphRamseyNumber 3 k)
    atTop atTop

theorem target : statement := sorry

end Statements.Erdos544RamseyGapDiverges
```

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