Note on the three-coloured bipartite Ramsey numbers for paths

Abstract

For bipartite graphs , the bipartite Ramsey number is the least positive integer p so that any coloring of the edges of with k colors will result in a copy of in the ith color for some i. In this paper, we investigate the appearance of simpler monochromatic graphs such as paths under a 3-colouring of the edges of a bipartite graph. we obtain the exact value of , and for , and for by a new method of proof.