Zarankiewicz’s Problem via ε-t-Nets
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.
Zarankiewicz’s Problem
ε-t-nets
pseudo-discs
VC-dimension
Theory of computation~Computational geometry
66:1-66:15
Regular Paper
https://arxiv.org/pdf/2311.13662.pdf
Chaya
Keller
Chaya Keller
Department of Computer Science, Ariel University, Israel
https://orcid.org/0000-0001-6400-3946
Research partially supported by Grant 1065/20 from the Israel Science Foundation.
Shakhar
Smorodinsky
Shakhar Smorodinsky
Department of Computer Science, Ben-Gurion University of the NEGEV, Beer Sheva, Israel
https://orcid.org/0000-0003-3038-6955
Research partially supported by Grant 1065/20 from the Israel Science Foundation.
10.4230/LIPIcs.SoCG.2024.66
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Chaya Keller and Shakhar Smorodinsky
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