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Erdős #564 · graph theory / ramsey theory

Let R3(n)R_3(n) be the minimal mm such that if the edges of the 33-uniform hypergraph on mm vertices are 22-coloured then there is a monochromatic copy of the complete 33-uniform hypergraph on nn vertices.

Is there s…

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 R3(n)R_3(n) be the minimal mm such that if the edges of the 33-uniform hypergraph on mm vertices are 22-coloured then there is a monochromatic copy of the complete 33-uniform hypergraph on nn vertices.

Is there some constant c>0c>0 such that

R3(n)≥22cn?R_3(n) \geq 2^{2^{cn}}?

Lean 4

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Formal statement

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Background

A special case of [562]. A problem of Erdős, Hajnal, and Rado [EHR65], who prove the bounds

2cn2<R3(n)<22n2^{cn^2}< R_3(n)< 2^{2^{n}}

for some constant c>0c>0.

Erdős, Hajnal, Máté, and Rado [EHMR84] have proved a doubly exponential lower bound for the corresponding problem with 44 colours.

This problem is #37 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 #564 link
  2. [2]Upstream Lean 4 statement for Erdős #564 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 564 link

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