Joshua Losh (University of Leeds) – The Zarankiewicz Problem

Abstract: Arising from extremal combinatorics, the global Zarankiewicz problem seeks an upper bound on the number of edges of a finite $r$-hypergraph where the edge relation is induced by some fixed hypergraph that does not contain the compete $r$-hypergraph $K_{k,\dots,k}$  as a subgraph for some $k$. Special cases of this problem, where the edge relation comes from a definable set in a particular structure, have been of interest to model theorists. This talk will give an introduction to the problem as well as some recent results.