The Ramsey number of mixed-parity cycles I

TL;DR

This paper determines the exact three-colour Ramsey number for large cycles of mixed parity, specifically when two are even and one is odd, improving previous asymptotic results to exact values.

Contribution

It provides an exact formula for the Ramsey number of three cycles with mixed parity for sufficiently large sizes, refining earlier asymptotic bounds.

Findings

Exact Ramsey number formula for large cycles of mixed parity

Improved from asymptotic to exact results

Self-contained proof for even longest cycle case

Abstract

Denote by $R(G_1, G_2, G_3)$ the minimum integer $N$ such that any three-colouring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of a graph $G_i$ coloured with colour $i$ for some $i\in{1,2,3}$. In a series of three papers of which this is the first, we consider the case where $G_1, G_2$ and $G_3$ are cycles of mixed parity. Specifically, in this and the subsequent paper, we consider $R(C_n,C_m,C_{\ell})$, where $n$ and $m$ are even and $\ell$ is odd. Figaj and \L uczak determined an asymptotic result for this case, which we improve upon to give an exact result. We prove that for $n,m$ and $\ell$ sufficiently large $R(C_n,C_m,C_\ell)=\max\{2n+m-3, n+2m-3, \tfrac{1}{2} n +\tfrac{1}{2} m + \ell - 2\}$. In the case that the longest cycle is of even length, the proof in this paper is self-contained. However, in the case that the longest cycle is of odd length, we require an additional technical result, the proof of which makes up the majority of the subsequent paper.