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OP-8868OpenCombinatorics

Erdős #162 · combinatorics / ramsey theory

Let α>0\alpha>0 and n≥1n\geq 1. Let F(n,α)F(n,\alpha) be the largest kk such that there exists some 2-colouring of the edges of KnK_n in which any induced subgraph HH on at least kk vertices contains more than $\alpha\binom…

MSC 05 · Combinatorics

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 α>0\alpha>0 and n≥1n\geq 1. Let F(n,α)F(n,\alpha) be the largest kk such that there exists some 2-colouring of the edges of KnK_n in which any induced subgraph HH on at least kk vertices contains more than α(∣H∣2)\alpha\binom{\lvert H\rvert}{2} many edges of each colour.

Prove that for every fixed 0≤α≤1/20\leq \alpha \leq 1/2, as n→∞n\to\infty,

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

for some constant cαc_\alpha.

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Background

It is easy to show with the probabilistic method that there exist c1(α),c2(α)c_1(\alpha),c_2(\alpha) such that

c1(α)log⁡n<F(n,α)<c2(α)log⁡n.c_1(\alpha)\log n < F(n,\alpha) < c_2(\alpha)\log n.

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 #162 link

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