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arXiv · 2510.17981

An improved upper bound for the multicolour Ramsey number of odd cycles

Maria Axenovich·Wouter Cames van Batenburg·Oliver Janzer·Lukas Michel·Mathieu Rundström
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Abstract

We show that the $k$-colour Ramsey number of an odd cycle of length $2 \ell + 1$ is at most $(4 \ell)^k \cdot k^{k/\ell}$. This proves a conjecture of Fox and is the first improvement in the exponent that goes beyond an absolute constant factor since the work of Bondy and Erdős from 1973.

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Maria Axenovich, Wouter Cames van Batenburg, Oliver Janzer, Lukas Michel, Mathieu Rundström. 2025-10-20. An improved upper bound for the multicolour Ramsey number of odd cycles. https://arxiv.org/abs/2510.17981

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