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{\'a}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{\'a}n's result to arbitrary graphs $H$ with $\chi(H) > 2$ (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.

翻译：极值图论的核心课题之一是研究函数$ex(n,H)$，该函数表示一个具有$n$个顶点的图在不包含固定图$H$作为子图的前提下所能拥有的最大边数。当$H$是$r$个顶点上的完全图时，Turán给出了完整的刻画。Erdős、Stone和Simonovits将Turán的结果推广到任意满足$\chi(H) > 2$（色数大于2）的图$H$。然而，对于二部图$H$，确定$ex(n, H)$的渐近性态仍然是一个广泛存在的开放性问题。该问题的经典实例是Zarankiewicz问题，即探究$ex(n, K_{t,t})$的渐近性态。本文综述了Zarankiewicz问题，重点关注几何中产生的图。关联几何尤其可视为Zarankiewicz问题在几何定义图中的具体表现。