Bipartite Ramsey numbers involving large $K_{n,n}$

Abstract

Let $br(H_1,H_2)$ be the bipartite Ramsey number for bipartite graphs $H_1$ and $H_2$. It is shown that the order of magnitude of $br(K_{t,n},K_{n,n})$ is $n^{t+1}/{(logn)}^t$ for $t\ge 1$ fixed and $n\to \infty$. Moreover, if $H$ is an isolate-free bipartite graph of order $h$ having bipartition $(A,B)$ that satisfies $\Delta \left(B\right)\le t$, then $br(H,K_{n,n})$ can be bounded from above by ${(hn/logn)}^t{(logn)}^{\alpha \left(t\right)}$ for large $n$, where $\alpha \left(1\right)=\alpha \left(2\right)=1$ and $\alpha \left(t\right)=0$ for $t\ge 3$.