Multicolor bipartite Ramsey numbers for paths, cycles, and stripes

Abstract

For the given bipartite graphs $G_1,G_2,\ldots ,G_t$, the multicolor bipartite Ramsey number $\textrm{BR}(G_1,G_2,\ldots ,G_t)$ is the smallest positive integer b, such that any t-edge-coloring of $K_{b,b}$ contains a monochromatic subgraph isomorphic to $G_i$ colored with the i-th color for some $1\le i\le t$. We compute the exact values of the bipartite Ramsey numbers $\textrm{BR}(P_i,C_{2n})$ and $\textrm{BR}(P_i,C_{2n},mK_2)$ for $i=3,5,7$ and $m,n\ge 2$, and $\textrm{BR}(P_a,P_b,mK_2)$ for all $a,b\ge 1$ and $m\ge 2$.