The Ramsey numbers R(C[sub m], K[sub 7]) and R(C[sub 7], K[sub 8])

Abstract

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 C[sub m] denote a cycle of length m and K[sub n] a complete graph of order n. In this paper we show that R(C[sub m], K[sub 7]) = 6m − 5 for m ≥ 7 and R(C[sub 7], K[sub 8]) = 43, with the former result confirming a conjecture due to Erdös, Faudree, Rousseau and Schelp that R(C[sub m], K[sub n]) = (m − 1)(n − 1)+ 1 for m ≥ n ≥ 3 and (m,n) ≠ (3,3) in the case where n = 7.