On Zarankiewicz's Problem for Intersection Hypergraphs of Geometric Objects
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866915489551220736 |
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| author | Chan, Timothy M. Keller, Chaya Smorodinsky, Shakhar |
| author_facet | Chan, Timothy M. Keller, Chaya Smorodinsky, Shakhar |
| contents | The hypergraph Zarankiewicz's problem, introduced by Erdős in 1964, asks for the maximum number of hyperedges in an $r$-partite hypergraph with $n$ vertices in each part that does not contain a copy of $K_{t,t,\ldots,t}$. Erdős obtained a near optimal bound of $O(n^{r-1/t^{r-1}})$ for general hypergraphs. In recent years, several works obtained improved bounds under various algebraic assumptions -- e.g., if the hypergraph is semialgebraic.
In this paper we study the 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 $\mathbb{R}^d$ and families of pseudo-discs. For axis-parallel boxes, we obtain the sharp bound $O_{d,r}(tn^{r-1}(\frac{\log n}{\log \log n})^{d-1})$. 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 $O_r(tn^{r-1}(\log n)^{r-2})$, 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 $\tildeΩ(n^{\frac{2r-2}{3r-2}})$.
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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_06490 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Zarankiewicz's Problem for Intersection Hypergraphs of Geometric Objects Chan, Timothy M. Keller, Chaya Smorodinsky, Shakhar Combinatorics Computational Geometry 05C35, 05C65 The hypergraph Zarankiewicz's problem, introduced by Erdős in 1964, asks for the maximum number of hyperedges in an $r$-partite hypergraph with $n$ vertices in each part that does not contain a copy of $K_{t,t,\ldots,t}$. Erdős obtained a near optimal bound of $O(n^{r-1/t^{r-1}})$ for general hypergraphs. In recent years, several works obtained improved bounds under various algebraic assumptions -- e.g., if the hypergraph is semialgebraic. In this paper we study the 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 $\mathbb{R}^d$ and families of pseudo-discs. For axis-parallel boxes, we obtain the sharp bound $O_{d,r}(tn^{r-1}(\frac{\log n}{\log \log n})^{d-1})$. 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 $O_r(tn^{r-1}(\log n)^{r-2})$, 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 $\tildeΩ(n^{\frac{2r-2}{3r-2}})$. 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. |
| title | On Zarankiewicz's Problem for Intersection Hypergraphs of Geometric Objects |
| topic | Combinatorics Computational Geometry 05C35, 05C65 |
| url | https://arxiv.org/abs/2412.06490 |