Generalized Ramsey--Turán density for cliques

Abstract

We study the generalized Ramsey--Turán function $\mathrm{RT}(n,K_s,K_t,o(n))$, which is the maximum possible number of copies of $K_s$ in an $n$-vertex $K_t$-free graph with independence number $o(n)$. The case when $s=2$ was settled by Erd{ő}s, S{ó}s, Bollob{á}s, Hajnal, and Szemerédi in the 1980s. We combinatorially resolve the general case for all $s\ge 3$, 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.