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OP-9067OpenGraph theory

Erdős #563 · graph theory / ramsey theory

Let F(n,α)F(n,\alpha) denote the smallest mm such that there exists a 22-colouring of the edges of KnK_n so that every X⊆[n]X\subseteq [n] with ∣X∣≥m\lvert X\rvert\geq m contains more than α(∣X∣2)\alpha \binom{\lvert X\rvert}{2} many…

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 F(n,α)F(n,\alpha) denote the smallest mm such that there exists a 22-colouring of the edges of KnK_n so that every X⊆[n]X\subseteq [n] with ∣X∣≥m\lvert X\rvert\geq m contains more than α(∣X∣2)\alpha \binom{\lvert X\rvert}{2} many edges of each colour.

Prove that, for every 0≤α<1/20\leq \alpha< 1/2,

F(n,α)∼cαlog⁡nF(n,\alpha)\sim c_\alpha\log n

for some constant cαc_\alpha depending only on α\alpha.

Lean 4

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Formal statement

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Background

It is easy to show via the probabilistic method that, for every 0≤α<1/20\leq \alpha<1/2,

F(n,α)≍αlog⁡n.F(n,\alpha)\asymp_\alpha \log n.

Note that when α=0\alpha=0 this is just asking for a 22-colouring of the edges of KnK_n which contains no monochromatic clique of size mm, and hence we recover the classical Ramsey numbers.

See also [161] for a generalisation to hypergraphs.

This problem is #39 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 #563 link
  2. [2]Formal Conjectures — Erdős Problem 563 link

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