Hypergraphs without complete partite subgraphs
Mubayi, Dhruv
Fix integers and and set . Let denote the complete -partite -uniform hypergraph with parts of size . We prove that the Zarankiewicz number provided . Previously this was known only for 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 , for example, it gives where the exponent 11/4 is optimal, whereas previously this was only known with 7 replaced by 721.