The number of cliques in hypergraphs with forbidden subgraphs

TL;DR

This paper investigates the maximum number of r-vertex cliques in (r-1)-uniform hypergraphs avoiding certain complete r-partite hypergraphs, establishing asymptotic bounds and connecting to known results in graph theory.

Contribution

It introduces new asymptotic bounds for clique counts in hypergraphs with forbidden subgraphs using the hypergraph removal lemma and Turán numbers, extending existing graph results.

Findings

Maximum clique count is o(n^{r - 1/(a_1...a_{r-1})})

Provides lower bounds via hypergraph Turán numbers

Extends graph results to hypergraph setting

Abstract

We study the maximum number of $r$-vertex cliques in $(r-1)$-uniform hypergraphs not containing complete $r$-partite hypergraphs $K_r^{(r-1)}(a_1, \dots, a_r)$. By using the hypergraph removal lemma, we show that this maximum is $o( n^{r - 1/(a_1 \cdots a_{r-1})} )$. This immediately implies the corresponding results of Mubayi and Mukherjee and of Balogh, Jiang, and Luo for graphs. We also provide a lower bound by using hypergraph Tur\'an numbers.