Erdős Problem #562 — open problem in combinatorics

Erdős Problem #562

Let R_r(n) denote the r-uniform hypergraph Ramsey number: the minimal m such that if we 2-colour all edges of the complete r-uniform hypergraph on m vertices then there must be some monochromatic copy of the complete r-uniform hypergraph on n vertices.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-562,
  title        = {Erdős Problem #562},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-562}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $R_{r} (n)$ denote the $r$ -uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$ -colour all edges of the complete $r$ -uniform hypergraph on $m$ vertices then there must be some monochromatic copy of the complete $r$ -uniform hypergraph on $n$ vertices.

Prove that, for $r \geq 3$ , $lo g_{r - 1} R_{r} (n) ≍_{r} n,$ where $lo g_{r - 1}$ denotes the $(r - 1)$ -fold iterated logarithm.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«562». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_562 : answer(sorry) ↔
    ∀ r ≥ 3, (fun n ↦ log^[r - 1] (hypergraphRamsey r n)) =Θ[atTop] (fun n ↦ (n : ℝ))

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/562. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/562

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.