On the Ramsey number R(Cn or Kn-1, Km) (m=3, 4)

The Ramsey number R(Cn or Kn1 , Km) is the smallest integer p such that every graph G on p vertices contains either a cycle en with length n or a K n 1 , or an independent set of order m. In this paper we prove that R(Cn or K n b K 3 ) = 2(n 2) + 1 (n 2: 5), R(Cn or K n 1 , K4) = 3(n 2) + 1 (n 2: 7). In particular, we prove that R(C4 or K 3 , K 3 ) = 6, R(C4 or K 3 , K4) = 8, R(C5 or K4, K 4 ) = 11 and R(C6 or K 5 , K 4) = 14.