Erdős Problem #563

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

Prove that, for every $0\leq \alpha< 1/2$,\[F(n,\alpha)\sim c_\alpha\log n\]for some constant $c_\alpha$ depending only on $\alpha$.

#563: [Er90b,p.21]

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It is easy to show via the probabilistic method that, for every $0\leq \alpha<1/2$,\[F(n,\alpha)\asymp_\alpha \log n.\]Note that when $\alpha=0$ this is just asking for a $2$-colouring of the edges of $K_n$ which contains no monochromatic clique of size $m$, 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.

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This page was last edited 18 January 2026.

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Additional thanks to: Boris Alexeev

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