A few remarks on Ramsey–Turán-type problems

Let H be a fixed forbidden graph and let f be a function of n. Denote by RT(n,H,f(n)) 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(n) independent vertices. The problem of estimating RT(n,H,f(n)) is one of the central questions of so-called Ramsey–Turán theory. ir 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.