On the maximum number of r-cliques in graphs free of complete r-partite subgraphs

We estimate the maximum possible number of cliques of size r in an n-vertex graph free of a fixed complete r-partite graph Kₒ䃑, ₒ䃒, , ₒ㶂. By viewing every r-clique as a hyperedge, the upper bound on the Tur\'an number of the complete r-partite hypergraphs gives the upper bound O (n^r - {1/₈=₁^{r-1sᵢ}}). We improve this to o (n^r - {1/₈=₁^{r-1sᵢ}}). The main tool in our proof is the graph removal lemma. We also provide several lower bound constructions.