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

TL;DR

This paper improves bounds on the maximum chromatic number of triangle-free graphs, confirming a conjecture and extending results to list chromatic number and edge-based bounds.

Contribution

It refines the upper bounds on chromatic and list chromatic numbers for triangle-free graphs, confirming a recent conjecture and providing tight bounds in terms of edges.

Findings

Improved the upper bound on the chromatic number by a factor of .

Extended bounds to list chromatic number, tight up to a constant.

Established tight bounds based on the number of edges.

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.