On s-Bipartite Ramsey Numbers Stars, Matchings and Double Stars

Abstract

For bipartite graphs $F$ and $H$ and a positive integer $s$, the $s$-bipartite Ramsey number $B{R}_s(F,H)$ of $F$ and $H$ is the smallest integer $t$ with $t\ge s$ such that every red-blue coloring of $K_{s,t}$ results in a red $F$ or a blue $H$. We evaluate this number for all positive integers $s$ when $F$ and $H$ are both stars, are both matchings, or one is a star and the other is a matching, as well as when $F=H$ is an arbitrary double star.