The Ramsey number of mixed-parity cycles III

TL;DR

This paper determines the exact three-color Ramsey number for large cycles of mixed parity, specifically when one cycle length is even and the others are odd, refining previous asymptotic results to exact values.

Contribution

It provides an exact formula for the Ramsey number of three mixed-parity cycles for large sizes, improving upon prior asymptotic bounds.

Findings

Exact Ramsey number formula for large cycles with mixed parity

Improved from asymptotic to exact results

Applicable for sufficiently large cycle lengths

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 third, we consider the case where $G_1, G_2$ and $G_3$ are cycles of mixed parity. Specifically, in this in this paper, we consider $R(C_n,C_m,C_{\ell})$, where $n$ is even and $m$ and $\ell$ are 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 $R(C_n,C_m,C_\ell)=\max\{4n-3, n+2m-3, n+2\ell-3\}$.