A survey of Zarankiewicz problems in geometry

Abstract

One of the central topics in extremal graph theory is the study of the function $ex(n,H)$, which represents the maximum number of edges a graph with $n$ vertices can have while avoiding a fixed graph $H$ as a subgraph. Tur{á}n provided a complete characterization for the case when $H$ is a complete graph on $r$ vertices. Erd{\H o}s, Stone, and Simonovits extended Tur{á}n's result to arbitrary graphs $H$ with $\ddot{I}‡(H)>2$ (chromatic number greater than 2). However, determining the asymptotics of $ex(n,H)$ for bipartite graphs $H$ remains a widely open problem. A classical example of this is Zarankiewicz's problem, which asks for the asymptotics of $ex(n,K_{t,t})$. In this paper, we survey Zarankiewicz's problem, with a focus on graphs that arise from geometry. Incidence geometry, in particular, can be viewed as a manifestation of Zarankiewicz's problem in geometrically defined graphs.