Zarankiewicz’s Problem via ϵ-t-Nets

Chaya Keller, Shakhar Smorodinsky

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 Kt,t. Kővári, Sós and Turán proved an upper bound of O(n2− 1t ). Fox et al. obtained an improved bound of O(n2− d1 ) 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(n2− d1 ) bound of Fox et al., and show improved bounds for several other classes of geometric intersection graphs, including a sharp O(n logloglognn ) bound for the intersection graph of two families of axis-parallel rectangles.

Original languageAmerican English
Title of host publication40th International Symposium on Computational Geometry, SoCG 2024
EditorsWolfgang Mulzer, Jeff M. Phillips
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959773164
DOIs
StatePublished - 1 Jun 2024
Event40th International Symposium on Computational Geometry, SoCG 2024 - Athens, Greece
Duration: 11 Jun 202414 Jun 2024

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume293

Conference

Conference40th International Symposium on Computational Geometry, SoCG 2024
Country/TerritoryGreece
CityAthens
Period11/06/2414/06/24

Keywords

  • VC-dimension
  • Zarankiewicz’s Problem
  • pseudo-discs
  • ϵ-t-nets

All Science Journal Classification (ASJC) codes

  • Software

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