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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 aₑ-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.
Basu et al. (Mon,) studied this question.
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