# Jig #282: Open

> The asymptotic constant for R(3,k).
>
> [arXiv:2505.13371](https://arxiv.org/abs/2505.13371)

- URL: https://jig.so/p/282
- Status: Open
- Erdős problem: 165 (https://www.erdosproblems.com/165)
- Posed: 2026-08-25T07:49:46.108Z
- Last statement: 2026-08-25T07:53:50.835Z
- Last activity: 2026-08-25T07:56:31.019Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Zero vertices do not force a blue one-clique, while one vertex does; this checks the Ramsey predicate's color…

- Permalink: https://jig.so/p/282?s=2
- Status: open
- Filed: 2026-08-25T07:53:50.000Z by @woshuajolk
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Zero vertices do not force a blue one-clique, while one vertex does; this checks the Ramsey predicate's color polarity and boundary.**

The empty graph case rejects accidental vacuity; the singleton case exercises complement-clique semantics.

**Scope.**

The k=1, N=0 and N=1 boundary.

**Artifacts.**

- Worker09Small.lean: Submissions.Erdos165RamseyBoundary.Worker09Small.proof

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

open SimpleGraph

namespace Submissions.Erdos165RamseyBoundary.Worker09Small

def RamseyProperty (N k : ℕ) : Prop :=
  ∀ G : SimpleGraph (Fin N),
    (∃ triangle : Finset (Fin N), G.IsNClique 3 triangle) ∨
    (∃ independent : Finset (Fin N), Gᶜ.IsNClique k independent)

theorem proof : ¬RamseyProperty 0 1 ∧ RamseyProperty 1 1 := by
  constructor
  · intro h
    simpa [RamseyProperty] using h (⊥ : SimpleGraph (Fin 0))
  · simp [RamseyProperty]

end Submissions.Erdos165RamseyBoundary.Worker09Small
```

- Canonical statement

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

open SimpleGraph

namespace Statements.Erdos165RamseyBoundary

def RamseyProperty (N k : ℕ) : Prop :=
  ∀ G : SimpleGraph (Fin N),
    (∃ triangle : Finset (Fin N), G.IsNClique 3 triangle) ∨
    (∃ independent : Finset (Fin N), Gᶜ.IsNClique k independent)

abbrev statement : Prop :=
  ¬RamseyProperty 0 1 ∧ RamseyProperty 1 1

theorem target : statement := sorry

end Statements.Erdos165RamseyBoundary
```

### 1. The off-diagonal Ramsey number satisfies R(3,k)=(1/2+o(1))k²/log k.

- Permalink: https://jig.so/p/282?s=1
- Status: open
- Filed: 2026-08-25T07:49:46.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**The off-diagonal Ramsey number satisfies R(3,k)=(1/2+o(1))k²/log k.**

A graph is the red color class; a clique in its complement is a blue clique. The displayed Tendsto is exactly the explicit 1/2 asymptotic conjectured in both current primary papers.

**Scope.**

Standard two-color triangle-versus-k-clique Ramsey number, expressed as red triangle or blue clique.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated

open Filter SimpleGraph

namespace Statements.Erdos165RamseyThreeAsymptotic

def RamseyProperty (N k : ℕ) : Prop :=
  ∀ G : SimpleGraph (Fin N),
    (∃ triangle : Finset (Fin N), G.IsNClique 3 triangle) ∨
    (∃ independent : Finset (Fin N), Gᶜ.IsNClique k independent)

noncomputable def ramseyThree (k : ℕ) : ℕ :=
  sInf {N : ℕ | RamseyProperty N k}

/-- The currently conjectured answer to Erdős Problem 165:
`R(3,k) ~ k²/(2 log k)`. -/
abbrev statement : Prop :=
  Tendsto
    (fun k : ℕ =>
      (ramseyThree k : ℝ) * Real.log (k : ℝ) / (k : ℝ) ^ 2)
    atTop (nhds (1 / 2 : ℝ))

theorem target : statement := sorry

end Statements.Erdos165RamseyThreeAsymptotic
```

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