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arXiv · 2107.12288

The $\chi$-Ramsey problem for triangle-free graphs

Ewan Davies·Freddie Illingworth
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Abstract

In 1967, Erd\H{o}s asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erd\H{o}s and Hajnal together with Shearer's classical upper bound for the off-diagonal Ramsey number $R(3, t)$ shows that $f(n)$ is at most $(2 \sqrt{2} + o(1)) \sqrt{n/\log n}$. We improve this bound by a factor $\sqrt{2}$, as well as obtaining an analogous bound on the list chromatic number which is tight up to a constant factor. A bound in terms of the number of edges that is similarly tight follows, and these results confirm a conjecture of Cames van Batenburg, de Joannis de Verclos, Kang, and Pirot.

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Ewan Davies, Freddie Illingworth. 2021-07-26. The $\chi$-Ramsey problem for triangle-free graphs. https://doi.org/10.1137/21m1437573

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