Zarankiewicz's problem via $ε$-t-nets
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910584875778048 |
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| author | Keller, Chaya Smorodinsky, Shakhar |
| author_facet | Keller, Chaya Smorodinsky, Shakhar |
| contents | 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 $K_{t,t}$. In one of the cornerstones of extremal graph theory, Kővári Sós and Turán proved an upper bound of $O(n^{2-\frac{1}{t}})$. In a celebrated result, Fox et al. obtained an improved bound of $O(n^{2-\frac{1}{d}})$ for graphs of VC-dimension $d$ (where $d<t$). Basit, Chernikov, Starchenko, Tao and Tran improved the bound for the case of semilinear graphs. At SODA'23, 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. We show that the existence of `small'-sized $ε$-t-nets implies upper bounds for Zarankiewicz's problem. 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 $O(n^{2-\frac{1}{d}})$ bound of Fox et al., and show improved bounds for several other classes of geometric intersection graphs, including a sharp $O(n\frac{\log n}{\log \log n})$ bound for the intersection graph of two families of axis-parallel rectangles. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2311_13662 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Zarankiewicz's problem via $ε$-t-nets Keller, Chaya Smorodinsky, Shakhar Combinatorics Computational Geometry 05C35 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 $K_{t,t}$. In one of the cornerstones of extremal graph theory, Kővári Sós and Turán proved an upper bound of $O(n^{2-\frac{1}{t}})$. In a celebrated result, Fox et al. obtained an improved bound of $O(n^{2-\frac{1}{d}})$ for graphs of VC-dimension $d$ (where $d<t$). Basit, Chernikov, Starchenko, Tao and Tran improved the bound for the case of semilinear graphs. At SODA'23, 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. We show that the existence of `small'-sized $ε$-t-nets implies upper bounds for Zarankiewicz's problem. 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 $O(n^{2-\frac{1}{d}})$ bound of Fox et al., and show improved bounds for several other classes of geometric intersection graphs, including a sharp $O(n\frac{\log n}{\log \log n})$ bound for the intersection graph of two families of axis-parallel rectangles. |
| title | Zarankiewicz's problem via $ε$-t-nets |
| topic | Combinatorics Computational Geometry 05C35 |
| url | https://arxiv.org/abs/2311.13662 |