On the Ramsey-Turán numbers of graphs and hypergraphs

Abstract: Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Tur\'an number of H, RT_t(n, H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G where f(n) is larger than the maximum number of vertices in a $K_t$-free induced subgraph of G. Erd\H{o}s, Hajnal, Simonovits, S\'os, and Szemer\'edi posed several open questions about RT_t(n,K_s,o(n)), among them finding the minimum s such that $RT_t(n,K_{t+s},o(n)) = \Omega(n^2)$, where it is easy to see that $RT_t(n,K_{t+1},o(n)) = o(n^2)$. In this paper, we answer this question by proving that $RT_t(n,K_{t+2},o(n)) = \Omega(n^2)$; our constructions also imply several results on the Ramsey-Tur\'an numbers of hypergraphs.