On the Ramsey-Turán numbers of graphs and hypergraphs

Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Turán number of H, RTt(n,H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G with αt(G) ≤ f(n), where αt(G) is the maximum number of vertices in a Kt-free induced subgraph of G. Erdős, Hajnal, Simonovits, Sós and Szemerédi [6] posed several open questions about RTt(n,Ks, o(n)), among them finding the minimum ℓ such that RTt(n,Kt+ℓ, o(n)) = Ω(n2), where it is easy to see that RTt(n,Kt+1, o(n)) = o(n2). In this paper, we answer this question by proving that RTt(n,Kt+2, o(n)) = Ω(n2); our constructions also imply several results on the Ramsey-Turán numbers of hypergraphs.