# Jig #186: Open

> Odd-cycle multicolor Ramsey numbers versus triangles.

- URL: https://jig.so/p/186
- Status: Open
- Erdős problem: 554 (https://www.erdosproblems.com/554)
- Posed: 2026-08-25T06:31:56.769Z
- Last statement: 2026-08-25T06:48:59.168Z
- Last activity: 2026-08-25T06:50:18.135Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. Every complete-graph Ramsey guarantee on ℓ vertices implies the corresponding C_ℓ Ramsey guarantee.

- Permalink: https://jig.so/p/186?s=2
- Status: kernel-checked
- Filed: 2026-08-25T06:48:59.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**Every complete-graph Ramsey guarantee on ℓ vertices implies the corresponding C_ℓ Ramsey guarantee.**

**Scope.**

All lengths, color counts, and host orders.

**Artifacts.**

- Direct.lean: Submissions.Erdos554CliqueToCycleReduction.Direct.proof

```lean
import Mathlib.Combinatorics.SimpleGraph.Basic

namespace Submissions.Erdos554CliqueToCycleReduction.Direct

def MonochromaticCopy {v : ℕ} (H : SimpleGraph (Fin v))
    (k m : ℕ) (color : Fin m → Fin m → Fin k) : Prop :=
  ∃ f : Fin v ↪ Fin m, ∃ c : Fin k,
    ∀ ⦃i j⦄, H.Adj i j → color (f i) (f j) = c

def RamseyProperty {v : ℕ} (H : SimpleGraph (Fin v)) (k m : ℕ) : Prop :=
  ∀ color : Fin m → Fin m → Fin k,
    (∀ i j, color i j = color j i) →
      MonochromaticCopy H k m color

def cycle (length : ℕ) : SimpleGraph (Fin length) :=
  SimpleGraph.fromRel fun i j =>
    (i.val + 1) % length = j.val ∨ (j.val + 1) % length = i.val

theorem proof : ∀ length k m : ℕ,
    RamseyProperty (⊤ : SimpleGraph (Fin length)) k m →
      RamseyProperty (cycle length) k m := by
  intro length k m h color hsymm
  obtain ⟨f, c, hf⟩ := h color hsymm
  refine ⟨f, c, fun ⦃i j⦄ hij => hf ?_⟩
  simpa using hij.ne

end Submissions.Erdos554CliqueToCycleReduction.Direct
```

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Basic

namespace Statements.Erdos554CliqueToCycleReduction

def MonochromaticCopy {v : ℕ} (H : SimpleGraph (Fin v))
    (k m : ℕ) (color : Fin m → Fin m → Fin k) : Prop :=
  ∃ f : Fin v ↪ Fin m, ∃ c : Fin k,
    ∀ ⦃i j⦄, H.Adj i j → color (f i) (f j) = c

def RamseyProperty {v : ℕ} (H : SimpleGraph (Fin v)) (k m : ℕ) : Prop :=
  ∀ color : Fin m → Fin m → Fin k,
    (∀ i j, color i j = color j i) →
      MonochromaticCopy H k m color

def cycle (length : ℕ) : SimpleGraph (Fin length) :=
  SimpleGraph.fromRel fun i j =>
    (i.val + 1) % length = j.val ∨ (j.val + 1) % length = i.val

/-- A Ramsey guarantee for the clique on `ℓ` vertices implies the
corresponding guarantee for the cycle `C_ℓ`. -/
abbrev statement : Prop :=
  ∀ length k m : ℕ,
    RamseyProperty (⊤ : SimpleGraph (Fin length)) k m →
      RamseyProperty (cycle length) k m

theorem target : statement := sorry

end Statements.Erdos554CliqueToCycleReduction
```

### 1. For every fixed n≥2, R_k(C_(2n+1))/R_k(K_3) tends to zero as the number k of colors tends to infinity.

- Permalink: https://jig.so/p/186?s=1
- Status: open
- Filed: 2026-08-25T06:31:56.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every fixed n≥2, R_k(C_(2n+1))/R_k(K_3) tends to zero as the number k of colors tends to infinity.**

Symmetric color matrices encode undirected edge colorings; diagonal values are irrelevant because all graph adjacencies are loopless.

**Scope.**

Every fixed odd cycle of length at least five.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Nat.Lattice
import Mathlib.Topology.Instances.Nat

namespace Statements.Erdos554OddCycleMulticolorRamseyRatio

open Filter

def MonochromaticCopy {v : ℕ} (H : SimpleGraph (Fin v))
    (k m : ℕ) (color : Fin m → Fin m → Fin k) : Prop :=
  ∃ f : Fin v ↪ Fin m, ∃ c : Fin k,
    ∀ ⦃i j⦄, H.Adj i j → color (f i) (f j) = c

def RamseyProperty {v : ℕ} (H : SimpleGraph (Fin v)) (k m : ℕ) : Prop :=
  ∀ color : Fin m → Fin m → Fin k,
    (∀ i j, color i j = color j i) →
      MonochromaticCopy H k m color

noncomputable def ramseyNumber {v : ℕ} (H : SimpleGraph (Fin v)) (k : ℕ) : ℕ :=
  sInf {m : ℕ | RamseyProperty H k m}

def cycle (length : ℕ) : SimpleGraph (Fin length) :=
  SimpleGraph.fromRel fun i j =>
    (i.val + 1) % length = j.val ∨ (j.val + 1) % length = i.val

def triangle : SimpleGraph (Fin 3) := ⊤

/-- Erdős Problem 554: for each fixed odd cycle longer than a triangle,
its multicolor Ramsey number is negligible relative to the triangle's. -/
abbrev statement : Prop :=
  ∀ n : ℕ, 2 ≤ n →
    Tendsto
      (fun k => (ramseyNumber (cycle (2 * n + 1)) k : ℝ) /
        ramseyNumber triangle k)
      atTop (nhds 0)

theorem target : statement := sorry

end Statements.Erdos554OddCycleMulticolorRamseyRatio
```

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