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 Kᵣ⁽ʳ⁻¹⁾(a₁, , aᵣ). By using the hypergraph removal lemma, we show that this maximum is o( n^r - 1/(a₁ ⋯ aᵣ₋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.