Hypergraphs without complete partite subgraphs

Fix integers r 2 and 1 s₁ sₑ-₁ t and set s=₈=₁^r-1sᵢ. Let K=K (s₁, , sₑ-₁, t) denote the complete r-partite r-uniform hypergraph with parts of size s₁, , sₑ-₁, t. We prove that the Zarankiewicz number z (n, K) = n^r-1/s-o (1) provided t> 3^s+o (s). Previously this was known only for t > ( (r-1) (s-1) ) ! due to Pohoata and Zakharov. Our novel approach, which uses Behrend's construction of sets with no 3 term arithmetic progression, also applies for small values of sᵢ, for example, it gives z (n, K (2, 2, 7) ) =n^11/4-o (1) where the exponent 11/4 is optimal, whereas previously this was only known with 7 replaced by 721.