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arXiv · 2512.16062

A note on the maximum ratio between chromatic number and clique number

Igor Araujo·Rafael Filipe·Rafael Miyazaki
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Abstract

Let $f(n)$ be the maximum, over all graphs $G$ on $n$ vertices, of the ratio $\frac{\chi(G)}{\omega(G)}$, where $\chi(G)$ denotes the chromatic number of $G$ and $\omega(G)$ the clique number of $G$. In 1967, Erd\H{o}s showed that \[ \Big( \frac{1}{4} +o(1) \Big) \frac{n}{(\log_2 n)^2} \le f(n) \le \big( 4+o(1) \big) \frac{n}{(\log_2 n)^2} .\] We show that \[ f(n) \le \big(c+o(1)\big) \frac{n}{(\log_2 n)^2}\] for some $c<3.72$. This follows from recent improvements in the asymptotics of Ramsey numbers and is the first improvement in the asymptotics of $f(n)$ established by Erd\H{o}s.

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Igor Araujo, Rafael Filipe, Rafael Miyazaki. 2025-12-18. A note on the maximum ratio between chromatic number and clique number. https://arxiv.org/abs/2512.16062

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