Another view of Bipartite Ramsey numbers

Abstract

For bipartite graphs $G$ and $H$ and a positive integer $m$, the $m$-bipartite Ramsey number $BR_m(G, H)$ 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 $BR_5(K_{2,2}, K_{3,3}) = BR_6(K_{2,2}, K_{3,3}) = 12$ and $BR_7(K_{2,2}, K_{3,3}) = BR_8(K_{2,2}, K_{3,3}) = 9$. In this article, by a short and easy argument we determine the exact value of $BR_m(K_{2,2}, K_{3,3})$ for each $m\geq 1$.