The Ramsey numbers R(Cm,K7) and R(C7,K8)

For two given graphs G1 and G2, the Ramsey number R(G1,G2) is the smallest integer n such that for any graph G of order n, either G contains G1 or the complement of G contains G2. Let Cm denote a cycle of length m and Kn a complete graph of order n. In this paper we show that R(Cm,K7)=6m−5 for m≥7 and R(C7,K8)=43, with the former result confirming a conjecture due to Erdös, Faudree, Rousseau and Schelp that R(Cm,Kn)=(m−1)(n−1)+1 for m≥n≥3 and (m,n)≠(3,3) in the case where n=7.