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

For two given graphs G(1) and G(2), the Ramsey number R(G(1), G(2)) is the smallest integer it such that for any graph G of order n, either G contains G(1) or the complement of G contains G(2). Let C-m denote a cycle of length in and K-n a complete graph of order n. In this paper we show that R(C-m, K-7) = 6m - 5 for m >= 7 and R(C-7, K-8) = 43, with the former result confirming a conjecture due to Erdos, Faudree, Rousseau and Schelp, that R(C,,,, Kn) = (m - 1)(n - 1) + 1 for m >= n >= 3 and (m, n) not equal (3, 3) in the case where n = 7. (c) 2007 Elsevier Ltd. All rights reserved.