Three colour bipartite Ramsey number of cycles and paths

TL;DR

This paper determines the asymptotic three-colour bipartite Ramsey numbers for paths and even cycles, advancing understanding of colourings in bipartite graphs and extending classical results.

Contribution

It provides the first asymptotic determination of the 3-colour bipartite Ramsey numbers for paths and even cycles.

Findings

Asymptotic values for 3-colour bipartite Ramsey numbers of paths.

Asymptotic values for 3-colour bipartite Ramsey numbers of even cycles.

Extension of classical bipartite Ramsey results to three colours.

Abstract

The $k$-colour bipartite Ramsey number of a bipartite graph $H$ is the least integer $n$ for which every $k$-edge-coloured complete bipartite graph $K_{n,n}$ contains a monochromatic copy of $H$. The study of bipartite Ramsey numbers was initiated, over 40 years ago, by Faudree and Schelp and, independently, by Gy\'arf\'as and Lehel, who determined the $2$-colour Ramsey number of paths. In this paper we determine asymptotically the $3$-colour bipartite Ramsey number of paths and (even) cycles.