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

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.

4 pages