A Note On the Ramsey Numbers $R(C_4,B_n)$

Abstract

The Ramsey number $R(C_4,B_n)$ is the smallest positive integer $m$ such that for every graph $F$ of order $m$, either $F$ contains $C_4$ (a quadrilateral) or $\overline{F}$ contains $B_n$ (a book graph $K_2+\overline{K_n}$ of order $n+2$). Previously, we computed $R(C_4,B_n)=n+9$ for $8\le n\le 12$. In this continuing work, we find that $R(C_4,B_{13})=22$ and surprisingly $R(C_4,B_{14})=24$, showing that their values are not incremented by one, as one might have suspected. The results are based on computer algorithms.