The cycle-complete graph Ramsey number r(C8,K8)

Abstract

The cycle-complete graph Ramsey number $r(C_m,K_n)$ 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 $r(C_m,K_n)=(m-1)(n-1)+1$ for all $m\ge n\ge 3$ (except $r(C_3,K_3)=6$). In this paper we will present a proof for the conjecture in the case $n=m=8$.