The hypergraph Zarankiewicz's problem, introduced by Erd\H{o}s in 1964, asks for the maximum number of hyperedges in an $r$-partite hypergraph with $n$ vertices in each part that does not contain a copy of $K_{t,t,\ldots,t}$. Erd\H{o}s obtained a near optimal bound of $O(n^{r-1/t^{r-1}})$ for general hypergraphs. In recent years, several works obtained improved bounds under various algebraic assumptions -- e.g., if the hypergraph is semialgebraic. In this paper we study the problem in a geometric setting -- for $r$-partite intersection hypergraphs of families of geometric objects. Our main results are essentially sharp bounds for families of axis-parallel boxes in $\mathbb{R}^d$ and families of pseudo-discs. For axis-parallel boxes, we obtain the sharp bound $O_{d,t}(n^{r-1}(\frac{\log n}{\log \log n})^{d-1})$. The best previous bound was larger by a factor of about $(\log n)^{d(2^{r-1}-2)}$. For pseudo-discs, we obtain the bound $O_t(n^{r-1}(\log n)^{r-2})$, which is sharp up to logarithmic factors. As this hypergraph has no algebraic structure, no improvement of Erd\H{o}s' 60-year-old $O(n^{r-1/t^{r-1}})$ bound was known for this setting. Futhermore, even in the special case of discs for which the semialgebraic structure can be used, our result improves the best known result by a factor of $\tilde{\Omega}(n^{\frac{2r-2}{3r-2}})$. To obtain our results, we use the recently improved results for the graph Zarankiewicz's problem in the corresponding settings, along with a variety of combinatorial and geometric techniques, including shallow cuttings, biclique covers, transversals, and planarity.

翻译：超图Zarankiewicz问题由Erdős于1964年提出，它要求确定一个不包含$K_{t,t,\ldots,t}$子图的$r$部超图（每部有$n$个顶点）所能包含的最大超边数。Erdős对一般超图得到了近乎最优的界$O(n^{r-1/t^{r-1}})$。近年来，多项研究在各种代数假设下（例如超图为半代数时）获得了改进的界。本文在几何背景下研究该问题——针对几何对象族构成的$r$部交超图。我们的主要结果得到了$\mathbb{R}^d$中轴平行箱族和伪圆盘族的本质尖锐界。对于轴平行箱族，我们得到了尖锐界$O_{d,t}(n^{r-1}(\frac{\log n}{\log \log n})^{d-1})$，该界较先前最佳结果缩小了约$(\log n)^{d(2^{r-1}-2)}$倍。对于伪圆盘族，我们得到界$O_t(n^{r-1}(\log n)^{r-2})$，该界在对数因子范围内是尖锐的。由于此类超图不具备代数结构，Erdős在60年前提出的$O(n^{r-1/t^{r-1}})$界在此背景下从未被改进。即使在可运用半代数结构的圆盘特例中，我们的结果也将已知最佳界改进了$\tilde{\Omega}(n^{\frac{2r-2}{3r-2}})$倍。为获得这些结果，我们运用了对应背景下图Zarankiewicz问题的最新改进结果，并结合了多种组合与几何技术，包括浅层切割、双团覆盖、横贯集以及平面性理论。