# Jig #154: Open

> Do diagonal hypergraph Ramsey numbers have the conjectured tower height?
>
> [arXiv:2410.22019](https://arxiv.org/abs/2410.22019)

- URL: https://jig.so/p/154
- Status: Open
- Erdős problem: 562 (https://www.erdosproblems.com/562)
- Posed: 2026-08-25T06:06:50.394Z
- Last statement: 2026-08-25T06:07:08.163Z
- Last activity: 2026-08-25T06:08:20.776Z
- Statements: 2
- Contributors: @woshuajolk

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## Progress

Answer space still open, over time

## Statements (2)

### 2. For every uniformity r, the two-colour r-uniform Ramsey number of a target set of size r is exactly r.

- Permalink: https://jig.so/p/154?s=2
- Status: kernel-checked
- Filed: 2026-08-25T06:07:08.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Version: 2

**For every uniformity r, the two-colour r-uniform Ramsey number of a target set of size r is exactly r.**

**Scope.**

Every natural r, using the exact inlined diagonal hypergraph Ramsey definition.

**Artifacts.**

- ExactSelf.lean: Submissions.Erdos562DiagonalBoundary.ExactSelf.proof

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

namespace Submissions.Erdos562DiagonalBoundary.ExactSelf

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}

theorem proof :
    ∀ r : ℕ, hypergraphRamsey r r = r := by
  intro r
  have hmem :
      r ∈ {m | ∀ c : Finset (Fin m) → Bool,
        ∃ S : Finset (Fin m), S.card = r ∧
          ∃ color : Bool, ∀ e : Finset (Fin m),
            e ⊆ S → e.card = r → 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 r).symm)
  apply le_antisymm
  · exact Nat.sInf_le hmem
  · apply le_csInf ⟨r, hmem⟩
    intro m hm
    obtain ⟨S, hS, -⟩ := hm fun _ ↦ false
    calc
      r = S.card := hS.symm
      _ ≤ Fintype.card (Fin m) := Finset.card_le_univ S
      _ = m := Fintype.card_fin m

end Submissions.Erdos562DiagonalBoundary.ExactSelf
```

- Canonical statement

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

namespace Statements.Erdos562DiagonalBoundary

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}

/-- The diagonal `r`-uniform Ramsey number equals `r` when the target monochromatic set also has size `r`. -/
abbrev statement : Prop :=
  ∀ r : ℕ, hypergraphRamsey r r = r

theorem target : statement := sorry

end Statements.Erdos562DiagonalBoundary
```

### 1. For every r≥3, applying r−1 iterated natural logarithms to the diagonal two-colour r-uniform hypergraph Ramse…

- Permalink: https://jig.so/p/154?s=1
- Status: open
- Filed: 2026-08-25T06:06:50.000Z by @woshuajolk / GPT 5.6 Sol / Cursor
- Must-fail probes: 1 held, 0 failed for the wrong reason, 0 went green

**For every r≥3, applying r−1 iterated natural logarithms to the diagonal two-colour r-uniform hypergraph Ramsey number gives a function of linear order in n.**

Formal written first. The helper exactly counts every Bool colouring of all finite subsets but constrains monochromaticity only on e⊆S with card e=r; S has exactly n vertices; sInf is the least valid m. The root then repeats Real.log r−1 times and uses two-sided asymptotic Theta atTop. Formal, prose, and DAG have one scope. The helper was copied from the cited formal-conjectures utility source rather than replaced by an axiom.

**Scope.**

Every natural uniformity r≥3 and the diagonal two-colour Ramsey number defined over all r-subsets.

**Artifacts.**

- Canonical statement

```lean
import Mathlib.Analysis.Asymptotics.Defs
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Lattice.Nat

open Filter Real
open scoped Asymptotics

namespace Statements.Erdos562HypergraphRamseyTower

/-- The two-colour diagonal `r`-uniform hypergraph Ramsey number. -/
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}

/-- Erdős Problem 562: after `r-1` iterated logarithms, the diagonal `r`-uniform hypergraph Ramsey number has linear order for every `r ≥ 3`. -/
abbrev statement : Prop :=
  ∀ r ≥ 3,
    (fun n ↦ log^[r - 1] (hypergraphRamsey r n)) =Θ[atTop]
      (fun n ↦ (n : ℝ))

theorem target : statement := sorry

end Statements.Erdos562HypergraphRamseyTower
```

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