A Note on the Multi-Colored Bipartite Ramsey Number for Paths

Abstract

For bipartite graphs [Formula: see text], the bipartite Ramsey number [Formula: see text] is the least positive integer [Formula: see text] so that any coloring of the edges of [Formula: see text] with [Formula: see text] colors will result in a copy of [Formula: see text] in the [Formula: see text]th color for some [Formula: see text]. In this paper, we get the exact value of [Formula: see text], and obtain the upper and lower bounds of [Formula: see text], where [Formula: see text] denotes a path with [Formula: see text] vertice s.