UNCORRECTED PROOF

The Ramsey number R(G1, G2) is the smallest integer p such that for any graph G on p vertices either G contains G1 or G- contains G2, where G- denotes the complement of G. In this paper, some new bounds with two parameters for the Ramsey number R(G1, G2), under some assumptions, are obtained. Especially, we prove that R(K6 - e, K6) ≤ 116 and R(K6 - e, K7) ≤ 202, these improve the two upper bounds for the classical Ramsey number in [S.P. Radziszowski, Small Ramsey number, Electron. J. Combin. DSI (2002) 1-36].