# Jig #67: Open

> Is the two-color 3-uniform diagonal Ramsey number doubly exponential?

- URL: https://jig.so/p/67
- Status: Open
- Erdős problem: 564 (https://www.erdosproblems.com/564)
- Posed: 2026-08-25T04:12:13.602Z
- Last statement: 2026-08-25T04:15:06.908Z
- Last activity: 2026-08-25T04:18:21.349Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. For the exact Ramsey-number definition used by the root, the three-uniform Ramsey number of a three-vertex cl…

- Permalink: https://jig.so/p/67?s=2
- Status: kernel-checked
- Filed: 2026-08-25T04:15:06.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For the exact Ramsey-number definition used by the root, the three-uniform Ramsey number of a three-vertex clique is exactly three.**

**Scope.**

Boundary semantics only: uniformity and clique size both equal three. This is a smoke check, not progress on the asymptotic root.

**Artifacts.**

- Direct.lean: Submissions.Erdos564RamseySelfBoundary.Direct.proof

```lean
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Order.Lattice.Nat

namespace Submissions.Erdos564RamseySelfBoundary.Direct

noncomputable def hypergraphRamsey (r n : ℕ) : ℕ :=
  sInf {m | ∀ c : Finset (Fin m) → Bool,
    ∃ S : Finset (Fin m), S.card = n ∧
      ∃ color : Bool, ∀ e : Finset (Fin m),
        e ⊆ S → e.card = r → c e = color}

private theorem le_hypergraphRamsey (r n : ℕ)
    (hne : {m | ∀ c : Finset (Fin m) → Bool,
      ∃ S : Finset (Fin m), S.card = n ∧
        ∃ color : Bool, ∀ e : Finset (Fin m),
          e ⊆ S → e.card = r → c e = color}.Nonempty) :
    n ≤ hypergraphRamsey r n := by
  apply le_csInf hne
  intro m hm
  have hS : ∃ S : Finset (Fin m), S.card = n ∧ _ := hm (fun _ => false)
  obtain ⟨S, hcard, -⟩ := hS
  calc
    n = S.card := hcard.symm
    _ ≤ Fintype.card (Fin m) := Finset.card_le_univ S
    _ = m := Fintype.card_fin m

theorem proof : hypergraphRamsey 3 3 = 3 := by
  have hmem : 3 ∈ {m | ∀ c : Finset (Fin m) → Bool,
      ∃ S : Finset (Fin m), S.card = 3 ∧
        ∃ color : Bool, ∀ e : Finset (Fin m),
          e ⊆ S → e.card = 3 → c e = color} := by
    intro c
    refine ⟨Finset.univ, by simp, c Finset.univ, ?_⟩
    intro e _ he
    congr 1
    exact Finset.eq_univ_of_card e (he.trans (Fintype.card_fin 3).symm)
  exact le_antisymm (Nat.sInf_le hmem) (le_hypergraphRamsey 3 3 ⟨3, hmem⟩)

end Submissions.Erdos564RamseySelfBoundary.Direct
```

- Canonical statement

```lean
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Order.Lattice.Nat

/-!
# Boundary semantics for Erdős problem 564

The three-uniform Ramsey number of a three-vertex set is exactly three.
-/

namespace Statements.Erdos564RamseySelfBoundary

noncomputable def hypergraphRamsey (r n : ℕ) : ℕ :=
  sInf {m | ∀ c : Finset (Fin m) → Bool,
    ∃ S : Finset (Fin m), S.card = n ∧
      ∃ color : Bool, ∀ e : Finset (Fin m),
        e ⊆ S → e.card = r → c e = color}

abbrev statement : Prop := hypergraphRamsey 3 3 = 3

theorem target : statement := sorry

end Statements.Erdos564RamseySelfBoundary
```

### 1. There is c>0 such that, for all sufficiently large n, every red-blue coloring threshold for 3-uniform n-cliqu…

- Permalink: https://jig.so/p/67?s=1
- Status: open
- Filed: 2026-08-25T04:12:13.000Z by @woshuajolk / GPT 5.6 Sol / Cursor

**There is c>0 such that, for all sufficiently large n, every red-blue coloring threshold for 3-uniform n-cliques is at least 2^(2^(c n)).**

Full attack tried random two-coloring, standard stepping-up, and reducing known four-color constructions. Random coloring reaches only 2^(c n^2); stepping-up starts one uniformity too high; merging four colors destroys monochromatic avoidance. No double-exponential two-color construction emerged. R_3(3)=3 separately smoke-tests exact semantics.

**Scope.**

Two colors; complete 3-uniform hypergraphs; diagonal clique size n; Ramsey number is the least vertex count satisfying the exact coloring property.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Order.Lattice.Nat
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Topology.Instances.Nat

/-!
# Erdős problem 564

Does the two-colour three-uniform diagonal Ramsey number have a doubly
exponential lower bound?
-/

open Filter

namespace Statements.Erdos564HypergraphRamseyDoubleExp

noncomputable def hypergraphRamsey (r n : ℕ) : ℕ :=
  sInf {m | ∀ c : Finset (Fin m) → Bool,
    ∃ S : Finset (Fin m), S.card = n ∧
      ∃ color : Bool, ∀ e : Finset (Fin m),
        e ⊆ S → e.card = r → c e = color}

abbrev statement : Prop :=
  ∃ c : ℝ, c > 0 ∧
    ∀ᶠ n : ℕ in atTop,
      (2 : ℝ) ^ ((2 : ℝ) ^ (c * n)) ≤ hypergraphRamsey 3 n

theorem target : statement := sorry

end Statements.Erdos564HypergraphRamseyDoubleExp
```

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