Many cliques with small degree powers

Abstract

Suppose $0 < p \le \infty$. For a simple graph $G$ with a vertex-degree sequence $d_1, \dots, d_n$ satisfying $(d_1^p + \dots + d_n^p)^{1/p} \le C$, we prove asymptotically sharp upper bounds on the number of $t$-cliques in $G$. This result bridges the $p = 1$ case, which is the notable Kruskal--Katona theorem, and the $p = \infty$ case, known as the Gan--Loh--Sudakov conjecture, and resolved by Chase. In particular, we demonstrate that the extremal construction exhibits a dichotomy between a single clique and multiple cliques at $p_0 = t - 1$. Our proof employs the entropy method.