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, RT_t(n,H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G with α_t(G) ≤ f(n), where α_t(G) is the maximum number of vertices in a K_t-free induced subgraph of G. Erdo{double acute}s, Hajnal, Simonovits, Sós and Szemerédi [6] posed several open questions about RT_t(n,K_s, o(n)), among them finding the minimum ℓ such that RT_t(n,K_t+ℓ, o(n)) = Ω(n²), where it is easy to see that RT_t(n,K_t+1, o(n)) = o(n²). In this paper, we answer this question by proving that RT_t(n,K_t+2, o(n)) = Ω(n²); our constructions also imply several results on the Ramsey-Turán numbers of hypergraphs.