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arXiv · 2212.06957

A random coloring process gives improved bounds for the Erd\H{o}s-Gy\'arf\'as problem on generalized Ramsey numbers

Patrick Bennett·Andrzej Dudek·Sean English
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Abstract

The Erd\H{o}s-Gy\'arf\'as number $f(n, p, q)$ is the smallest number of colors needed to color the edges of the complete graph $K_n$ so that all of its $p$-clique spans at least $q$ colors. In this paper we improve the best known upper bound on $f(n, p, q)$ for many fixed values of $p, q$ and large $n$. Our proof uses a randomized coloring process, which we analyze using the so-called differential equation method to establish dynamic concentration.

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Patrick Bennett, Andrzej Dudek, Sean English. 2022-12-14. A random coloring process gives improved bounds for the Erd\H{o}s-Gy\'arf\'as problem on generalized Ramsey numbers. https://arxiv.org/abs/2212.06957

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