On Some Ramsey Numbers of $C_4$ versus $K_{2,n}$

Abstract

For given graphs $H_1$ and $H_2$, the $Ramseynumber$ $R(H_1,H_2)$ is the smallest positive integer $n$ such that if we arbitrarily color the edges of the complete graph $K_n$ with two colors, 1 (red) and 2 (blue), then there is a monochromatic copy of $H_1$ colored with 1 or $H_2$ colored with 2.

We show that if $n$ is even, $q=\lceil \sqrt{n}\rceil$ is odd, and $s=n–(q-1{)}^2\le \frac{q}{2}$, then $R(K_{2,2},K_{2,n})\le n+2q–1$, where $K_{n,m}$ are complete bipartite graphs. This bound provides the exact value of $R(K_{2,2},K_{2,18})=27$. Moreover, we show that $R(K_{2,2},K_{2,14})=22$ and $R(K_{2,2},K_{2,15})=24$.