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.