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On Zarankiewicz’s Problem for Intersection Hypergraphs of Geometric Objects

Authors Timothy M. Chan , Chaya Keller , Shakhar Smorodinsky

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  • Theory of computation → Computational geometry
Keywords
  • Zarankiewicz’s Problem
  • hypergraphs
  • intersection graphs
  • axis-parallel boxes
  • pseudo-discs

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Abstract

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 ℝ^d and families of pseudo-discs. For axis-parallel boxes, we obtain the sharp bound O_{d,t}(n^{r-1}((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 Erdős' 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 Ω̃(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.

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Timothy M. Chan, Chaya Keller, and Shakhar Smorodinsky. On Zarankiewicz’s Problem for Intersection Hypergraphs of Geometric Objects. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 33:1-33:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.33

Author Details

Timothy M. Chan
  • Siebel School of Computing and Data Science, University of Illinois at Urbana-Champaign, Urbana, IL, USA
Chaya Keller
  • School of Computer Science, Ariel University, Israel
Shakhar Smorodinsky
  • Department of Computer Science, Ben-Gurion University of the NEGEV, Be'er Sheva, Israel

Funding

  • Chan, Timothy M.: Supported by NSF Grant CCF-2224271.
  • Keller, Chaya: Partially supported by Grant 1065/20 from the Israel Science Foundation.
  • Smorodinsky, Shakhar: Partially supported by Grant 1065/20 from the Israel Science Foundation.

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