The $\chi$-Ramsey Problem for Triangle-Free Graphs
Abstract
In 1967, Erdös asked for the greatest chromatic number, $f(n)$, amongst all $n$-vertex, triangle-free graphs. An observation of Erdös 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 et al. [Electron. J. Combin., 27 (2020), P2.34].
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Published In
SIAM Journal on Discrete Mathematics
Pages: 1124 - 1134
DOI: 10.1137/21M1437573
ISSN (online): 1095-7146
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© 2022, Society for Industrial and Applied Mathematics.
History
Submitted: 2 August 2021
Accepted: 27 January 2022
Published online: 28 April 2022
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