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arXiv · 1901.05834

Multicolour bipartite Ramsey number of paths

Matija Bucic·Shoham Letzter·Benny Sudakov
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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.

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Matija Bucic, Shoham Letzter, Benny Sudakov. 2019-01-17. Multicolour bipartite Ramsey number of paths. https://arxiv.org/abs/1901.05834

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