Generalized Ramsey--Tur\'an density for cliques

We study the generalized Ramsey--Tur\'an function RT (n, Kₛ, Kₜ, o (n) ), which is the maximum possible number of copies of Kₛ in an n-vertex Kₜ-free graph with independence number o (n). The case when s=2 was settled by Erdos, S\'os, Bollob\'as, Hajnal, and Szemer\'edi in the 1980s. We combinatorially resolve the general case for all s 3, showing that the (asymptotic) extremal graphs for this problem have simple (bounded) structures. In particular, it implies that the extremal structures follow a periodic pattern when t is much larger than s. Our results disprove a conjecture of Balogh, Liu, and Sharifzadeh and show that a relaxed version does hold.