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

We show that the k -colour Ramsey number of an odd cycle of length 2 ℓ + 1 is at most ( 4 ℓ ) k ⋅ k k / ℓ . 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.