The Ramsey numbers and

For two given graphs G 1 and G 2 , the Ramsey number R ( G 1 , G 2 ) is the smallest integer n 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 m and K n a complete graph of order n . In this paper we show that R ( C m , K 7 ) = 6 m − 5 for m ≥ 7 and R ( C 7 , K 8 ) = 43 , with the former result confirming a conjecture due to Erdös, Faudree, Rousseau and Schelp that R ( C m , K n ) = ( m − 1 ) ( n − 1 ) + 1 for m ≥ n ≥ 3 and ( m , n ) ≠ ( 3 , 3 ) in the case where n = 7 .