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 Erd{{o}}s, S{\'o}s, Bollob{\'a}s, Hajnal, and Szemer\'{e}di 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.