Big line or big convex polygon
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arXiv
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| Hauptverfasser: | , , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911868477505536 |
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| author | Conlon, David Fox, Jacob He, Xiaoyu Mubayi, Dhruv Suk, Andrew Verstraete, Jacques |
| author_facet | Conlon, David Fox, Jacob He, Xiaoyu Mubayi, Dhruv Suk, Andrew Verstraete, Jacques |
| contents | Let $ES_{\ell}(n)$ be the minimum $N$ such that every $N$-element point set in the plane contains either $\ell$ collinear members or $n$ points in convex position. We prove that there is a constant $C>0$ such that, for each $\ell, n \ge 3$,
$$ (3\ell - 1) \cdot 2^{n-5} < ES_{\ell}(n) < \ell^2 \cdot 2^{n+ C\sqrt{n\log n}}.$$ A similar extension of the well-known Erd\H os--Szekeres cups-caps theorem is also proved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03455 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Big line or big convex polygon Conlon, David Fox, Jacob He, Xiaoyu Mubayi, Dhruv Suk, Andrew Verstraete, Jacques Combinatorics Let $ES_{\ell}(n)$ be the minimum $N$ such that every $N$-element point set in the plane contains either $\ell$ collinear members or $n$ points in convex position. We prove that there is a constant $C>0$ such that, for each $\ell, n \ge 3$, $$ (3\ell - 1) \cdot 2^{n-5} < ES_{\ell}(n) < \ell^2 \cdot 2^{n+ C\sqrt{n\log n}}.$$ A similar extension of the well-known Erd\H os--Szekeres cups-caps theorem is also proved. |
| title | Big line or big convex polygon |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2405.03455 |