The number of cliques in hypergraphs with forbidden subgraphs

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 $o(n^{r-1\mathrm{/}(a_1\cdots a_{r-1})})$. 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 Turán numbers.