Multicolour bipartite Ramsey number of paths

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,K-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 Gyarfas 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, Gyarfas, Krueger, Ruszinko, and Sarkozy, 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.