Size bipartite Ramsey numbers

Abstract

Let $B$, $B_1$ and $B_2$ be bipartite graphs, and let $B\to (B_1,B_2)$ signify that any red–blue edge-coloring of $B$ contains either a red $B_1$ or a blue $B_2$. The size bipartite Ramsey number $\widehat{br}(B_1,B_2)$ is defined as the minimum number of edges of a bipartite graph $B$ such that $B\to (B_1,B_2)$. It is shown that $\widehat{br}(K_{m,n},K_{m,n})$ is linear on $n$ with $m$ fixed, and $\widehat{br}(K_{n,n},K_{n,n})$ is between $c_1{n}^2{2}^n$ and $c_2{n}^3{2}^n$ for some positive constants $c_1$ and $c_2$.