On the Ramsey-Turán number with small s-independence number

Let s be an integer, f = f(n) a function, and H a graph. Define the Ramsey-Turán number RTs(n,H, f) as the maximum number of edges in an H-free graph G of order n with αs(G) < f , where αs(G) is the maximum number of vertices in a Ks-free induced subgraph of G. The Ramsey-Turán number attracted a considerable amount of attention and has been mainly studied for f not too much smaller than n. In this paper we consider RTs(n,Kt, n ) for fixed δ < 1. We show that for an arbitrarily small ε > 0 and 1/2 < δ < 1, RTs(n,Ks+1, n ) = Ω(n1+δ−ε) for all sufficiently large s. This is nearly optimal, since a trivial upper bound yields RTs(n,Ks+1, n ) = O(n). Furthermore, the range of δ is as large as possible. We also consider more general cases and find bounds on RTs(n,Ks+r, n ) for fixed r ≥ 2. Finally, we discuss a phase transition of RTs(n,K2s+1, f) extending some recent result of Balogh, Hu and Simonovits.