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 Ks₁, s₂, …, sᵣ. 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/∏ᵢ₌₁ʳ⁻¹sᵢ). We improve this to o(n^r - 1/∏ᵢ₌₁ʳ⁻¹sᵢ). The main tool in our proof is the graph removal lemma. We also provide several lower bound constructions.