A note on Ramsey numbers with two parameters

The Ramsey number R ( G 1 , G 2 ) is the smallest integer p such that for any graph G on p vertices either G contains G 1 or G ¯ contains G 2 , where G ¯ denotes the complement of G . In this paper, some new bounds with two parameters for the Ramsey number R ( G 1 , G 2 ) , under some assumptions, are obtained. Especially, we prove that R ( K 6 − e , K 6 ) ≤ 116 and R ( K 6 − e , K 7 ) ≤ 202 , these improve the two upper bounds for the classical Ramsey number in [S.P. Radziszowski, Small Ramsey number, Electron. J. Combin. DS1 (2002) 1–36].