A few remarks on Ramsey-Tura'n-type problems

Let H be a fixed forbidden graph and let f be a function of n: Denote by RTðn;H; f ðnÞÞ the maximum number of edges a graph G on n vertices can have without containing H as a subgraph and also without having at least f ðnÞ independent vertices. The problem of estimating RTðn;H; f ðnÞÞ is one of the central questions of so-called Ramsey–Turán theory. In their recent paper (Discrete Math. 229 (2001) 293–340), Simonovits and Sós gave an excellent survey of this theory and mentioned some old and new interesting open questions. In this short paper we obtain some new bounds for Ramsey–Turán-type problems. These results give partial answers for some of the questions. r 2002 Elsevier Science (USA). All rights reserved.