Hypergraphs without complete partite subgraphs

Fix integers r ≥ 2 and 1≤ s₁≤ ⋯ ≤ sᵣ₋₁≤ t and set s=∏ ᵢ₌₁ʳ⁻¹sᵢ . Let K=K(s₁, … , sᵣ₋₁, t) denote the complete r -partite r -uniform hypergraph with parts of size s₁, … , sᵣ₋₁, t . We prove that the Zarankiewicz number z(n, K)= nr-1/s-o(1) provided t> 3ˢ⁺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))=n11/4-o(1) where the exponent 11/4 is optimal, whereas previously this was only known with 7 replaced by 721.