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Zarankiewicz’s Problem via ε-t-Nets

Authors Chaya Keller , Shakhar Smorodinsky

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Author Details

Chaya Keller
  • Department of Computer Science, Ariel University, Israel
Shakhar Smorodinsky
  • Department of Computer Science, Ben-Gurion University of the NEGEV, Beer Sheva, Israel

Funding

  • Keller, Chaya: Research partially supported by Grant 1065/20 from the Israel Science Foundation.
  • Smorodinsky, Shakhar: Research partially supported by Grant 1065/20 from the Israel Science Foundation.

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Chaya Keller and Shakhar Smorodinsky. Zarankiewicz’s Problem via ε-t-Nets. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 66:1-66:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.66

Abstract

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}. Kővári, Sós and Turán proved an upper bound of O(n^{2-1/t}). Fox et al. obtained an improved bound of O(n^{2-1/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. 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 ε-t-nets - a recently introduced generalization of the classical notion of ε-nets. 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^{2-1/d}) bound of Fox et al., and show improved bounds for several other classes of geometric intersection graphs, including a sharp O(n {log n}/{log log n}) bound for the intersection graph of two families of axis-parallel rectangles.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Zarankiewicz’s Problem
  • ε-t-nets
  • pseudo-discs
  • VC-dimension

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