# Multicolour bipartite Ramsey number of paths

## TL;DR

This paper determines the asymptotic 4-colour bipartite Ramsey number for paths and cycles, advancing understanding of colourings in bipartite graphs and providing near-tight bounds for multiple colours.

## Contribution

It extends previous work by asymptotically determining the 4-colour bipartite Ramsey number of paths and cycles, and offers new upper bounds for general k-colour cases.

## Key findings

- Asymptotic 4-colour bipartite Ramsey number of paths and cycles determined.
- New upper bounds for k-colour bipartite Ramsey numbers close to tight.
- Progress on bipartite Ramsey numbers for multiple 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 bipartite Ramsey number of paths. Recently the $3$-colour Ramsey number of paths and (even) cycles, was essentially determined as well. Improving the results of DeBiasio, Gy\'arf\'as, Krueger, Ruszink\'o, and S\'ark\"ozy, in this paper we determine asymptotically the $4$-colour bipartite Ramsey number of paths and cycles. We also provide new upper bounds on the $k$-colour bipartite Ramsey numbers of paths and cycles which are close to being tight.