Generalized Ramsey-Tur\'an Numbers

The Ramsey-Tur\'an problem for Kₚ asks for the maximum number of edges in an n-vertex Kₚ-free graph with independence number $o(n)$. In a natural generalization of the problem, cliques larger than the edge K₂ are counted. Let { RT}(n,#Kq,Kₚ,o(n)) denote the maximum number of copies of Kq in an n-vertex Kₚ-free graph with independence number $o(n)$. Balogh, Liu and Sharifzadeh determined the asymptotics of { RT}(n,# K₃,Kₚ,o(n)). In this paper we will establish the asymptotics for counting copies of K₄, K₅, and for the case p ≥ 5q. We also provide a family of counterexamples to a conjecture of Balogh, Liu and Sharifzadeh.