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Erdős #562 · graph theory / ramsey theory

Let Rr(n)R_r(n) denote the rr-uniform hypergraph Ramsey number: the minimal mm such that if we 22-colour all edges of the complete rr-uniform hypergraph on mm vertices then there must be some monochromatic copy of the …

MSC 05C · Graph theory

Inferred from the community field; finer classification has not been reviewed.

Source reports OPENAs of 2026-09-08Source & evidence ↓
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Problem statement

Let Rr(n)R_r(n) denote the rr-uniform hypergraph Ramsey number: the minimal mm such that if we 22-colour all edges of the complete rr-uniform hypergraph on mm vertices then there must be some monochromatic copy of the complete rr-uniform hypergraph on nn vertices.

Prove that, for r≥3r\geq 3,

log⁡r−1Rr(n)≍rn,\log_{r-1} R_r(n) \asymp_r n,

where log⁡r−1\log_{r-1} denotes the (r−1)(r-1)-fold iterated logarithm. That is, does Rr(n)R_r(n) grow like

22⋯n2^{2^{\cdots n}}

where the tower of exponentials has height r−1r-1?

Lean 4

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Formal statement

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Background

A problem of Erdős, Hajnal, and Rado [EHR65]. A generalisation of [564].

This problem is #38 in Ramsey Theory in the graphs problem collection.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #562 link
  2. [2]Upstream Lean 4 statement for Erdős #562 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 562 link

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