The cycle-complete graph Ramsey number r(C6,K8)≤38

Abstract

The cycle-complete graph Ramsey number $r\text{(}{C}_m\text{, }{K}_n\text{)}$ is the smallest integer $N$ such that every graph $G$ of order $N$ contains a cycle $C_m$ on m vertices or has independent number $\alpha (G)\ge n$. It has been conjectured by Erdős, Faudree, Rousseau and Schelp that $r\text{(}{C}_m\text{,}{K}_n\text{)=(}m-1)(n-1)+1$ for all $m\ge n\ge 3$ (except $r(C_3,K_3)=6$). In this paper, we show that $r(C_6,K_8)\le 38$.