Multicolour Ramsey Numbers of Odd Cycles

TL;DR

This paper constructs specific edge colourings of complete graphs to establish new lower bounds on multicolour Ramsey numbers of odd cycles, advancing understanding of these combinatorial parameters.

Contribution

It introduces a method to colour complete graphs avoiding small monochromatic odd cycles, providing new lower bounds for multicolour Ramsey numbers of odd cycles.

Findings

Existence of colourings with no small monochromatic odd cycles

New lower bounds for multicolour Ramsey numbers of odd cycles

Progress on longstanding problems by Erd ext{"o}s, Graham, and Chung

Abstract

We show that for any positive integer $r$ there exists an integer $k$ and a $k$-colouring of the edges of $K_{2^{k}+1}$ with no monochromatic odd cycle of length less than $r$. This makes progress on a problem of Erd\H{o}s and Graham and answers a question of Chung. We use these colourings to give new lower bounds on the $k$-colour Ramsey number of the odd cycle and prove that, for all odd $r$ and all $k$ sufficiently large, there exists a constant $\epsilon = \epsilon(r) > 0$ such that $R_{k}(C_{r}) > (r-1)(2+\epsilon)^{k-1}$.