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

TL;DR

This paper establishes a new upper bound for the multicolour Ramsey number of odd cycles, improving upon previous results and confirming a conjecture by Fox, marking a significant advancement in combinatorial graph theory.

Contribution

The paper proves a tighter upper bound for the multicolour Ramsey number of odd cycles, advancing the understanding of Ramsey theory and confirming a longstanding conjecture.

Findings

New upper bound: $(4 \, \ell)^k \cdot k^{k/\ell}$ for the multicolour Ramsey number of odd cycles

First improvement in the exponent beyond a constant factor since 1973

Confirms a conjecture of Fox in the field of combinatorics

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 Erd\H{o}s from 1973.