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arXiv · 2607.16118

On the Erdős-Rogers function

Robert Morris·Julian Sahasrabudhe·Jacques Verstraëte
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Abstract

We show that the Erdős-Rogers function $f_{s,s+1}(n)$ satisfies $$f_{s,s+1}(n) = Θ( \sqrt{n \log n} )$$ for every $s \ge 2$. More precisely, we construct a $K_{s+1}$-free graph on $n$ vertices in which every set of at least $C(s)\sqrt{n \log n}$ vertices contains a copy of $K_s$ for some constant $C(s)$, which implies the upper bound. The matching lower bound follows from a theorem of Joret, Micek, Reed and Smid on the clique chromatic number of a graph.

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Robert Morris, Julian Sahasrabudhe, Jacques Verstraëte. 2026-07-17. On the Erdős-Rogers function. https://arxiv.org/abs/2607.16118

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