Another view of Bipartite Ramsey numbers

Abstract

For bipartite graphs $G$ and $H$ and a positive integer $m$, the $m$-bipartite Ramsey number $B{R}_m(G,H)$ of $G$ and $H$ is the smallest integer $n$, such that every red-blue coloring of $K_{m,n}$ results in a red $G$ or a blue $H$. Zhenming Bi, Gary Chartrand and Ping Zhang in \cite{bi2018another} evaluate this numbers for all positive integers $m$ when $G=K_{2,2}$ and $H\in \{K_{2,3},K_{3,3}\}$, especially in a long and hard argument they showed that $B{R}_5(K_{2,2},K_{3,3})=B{R}_6(K_{2,2},K_{3,3})=12$ and $B{R}_7(K_{2,2},K_{3,3})=B{R}_8(K_{2,2},K_{3,3})=9$. In this article, by a short and easy argument we determine the exact value of $B{R}_m(K_{2,2},K_{3,3})$ for each $m\ge 1$.