On Zarankiewicz's Problem for Intersection Hypergraphs of Geometric Objects

In this paper we study the hypergraph Zarankiewicz's 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 Rd and families of pseudo-discs. For axis-parallel boxes, we obtain the sharp bound Od,t(nr-1(log n/log log n)d-1). The best previous bound was larger by a factor of about (log n)d(2r-1-2). For pseudo-discs, we obtain the bound Ot(nr-1(log n)r-2), which is sharp up to logarithmic factors. As this hypergraph has no algebraic structure, no improvement of Erdos' 60-year-old O(nr-(1/tr-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 Ω (n 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.