The classical Zarankiewicz's problem asks for the maximum number of edges in a bipartite graph on $n$ vertices which does not contain the complete bipartite graph $K_{t,t}$. In one of the cornerstones of extremal graph theory, K\H{o}v\'ari S\'os and Tur\'an proved an upper bound of $O(n^{2-\frac{1}{t}})$. In a celebrated result, Fox et al. obtained an improved bound of $O(n^{2-\frac{1}{d}})$ for graphs of VC-dimension $d$ (where $d<t$). Basit, Chernikov, Starchenko, Tao and Tran improved the bound for the case of semilinear graphs. At SODA'23, Chan and Har-Peled further improved Basit et al.'s bounds and presented (quasi-)linear upper bounds for several classes of geometrically-defined incidence graphs, including a bound of $O(n \log \log n)$ for the incidence graph of points and pseudo-discs in the plane. In this paper we present a new approach to Zarankiewicz's problem, via $\epsilon$-t-nets - a recently introduced generalization of the classical notion of $\epsilon$-nets. We show that the existence of `small'-sized $\epsilon$-t-nets implies upper bounds for Zarankiewicz's problem. Using the new approach, we obtain a sharp bound of $O(n)$ for the intersection graph of two families of pseudo-discs, thus both improving and generalizing the result of Chan and Har-Peled from incidence graphs to intersection graphs. We also obtain a short proof of the $O(n^{2-\frac{1}{d}})$ bound of Fox et al., and show improved bounds for several other classes of geometric intersection graphs, including a sharp $O(n\frac{\log n}{\log \log n})$ bound for the intersection graph of two families of axis-parallel rectangles.

翻译：经典的 Zarankiewicz 问题旨在寻找不含完全二分图 $K_{t,t}$ 的 $n$ 顶点二分图的最大边数。在图极值理论的里程碑式工作中，Kővári、Sós 和 Turán 证明了该问题的上界为 $O(n^{2-\frac{1}{t}})$。Fox 等人通过引入 VC 维数 $d$（其中 $d<t$）的图，将上界改进为 $O(n^{2-\frac{1}{d}})$。Basit、Chernikov、Starchenko、Tao 和 Tran 进一步改进了半线性图的上界。在 SODA'23 会议上，Chan 和 Har-Peled 改进了 Basit 等人的结果，并给出了若干几何定义关联图的（拟）线性上界，包括平面中点与伪圆盘关联图的 $O(n \log \log n)$ 上界。本文通过 $\varepsilon$-$t$-网（经典 $\varepsilon$-网概念的新型推广）提出了一种解决 Zarankiewicz 问题的新方法。我们证明：存在“小规模”的 $\varepsilon$-$t$-网可导出 Zarankiewicz 问题的上界。利用这一新方法，我们得到两族伪圆盘交图的紧界 $O(n)$，从而改进了 Chan 和 Har-Peled 的结果，并将其从关联图推广至交图。此外，我们给出了 Fox 等人 $O(n^{2-\frac{1}{d}})$ 界的一个简短证明，并展示了其他几类几何交图的改进上界，包括两族轴平行矩形交图的紧界 $O(n\frac{\log n}{\log \log n})$。