R(C6, K5) = 21andR(C7, K5) = 25

The Ramsey number R(Cn, Km) is the smallest integer p such that any graph G on p vertices either contains a cycleCn with length n or contains an independent set with order m. In this paper we prove that R(Cn, K5) = 4(n − 1) + 1 (n = 6, 7).