A survey of Zarankiewicz problems in geometry
Shakhar Smorodinsky
TL;DR
This survey synthesizes results on Zarankiewicz-type extremal problems with a geometric lens, focusing on ex(n,H) for bipartite H and the special roles of incidence and semi-algebraic graphs. It surveys foundational bounds (Turán, Erdős–Stone–Simonovits, KST) and progresses achieved under structural constraints such as forbidding induced subgraphs and bounding VC-dimension, including eps-net methods and polynomial partitioning. It highlights major geometric advances: semi-algebraic graphs achieving subquadratic bounds, extended Szemerédi–Trotter bounds, and linear bounds for pseudo-discs and string graphs, as well as hypergraph extensions with recent epsilon-free bounds. The article also outlines open problems and conjectures, pointing to deeper interactions between extremal combinatorics, incidence geometry, and computational geometry with broad theoretical significance and potential applications.
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 $χ(H) > 2$ (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.