Excursions in Sylvester-Gallai land
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866912768703070208 |
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| author | Barany, Imre Du, Julia Q. Schwarz, Dan Yuan, Liping Zamfirescu, Tudor |
| author_facet | Barany, Imre Du, Julia Q. Schwarz, Dan Yuan, Liping Zamfirescu, Tudor |
| contents | The Sylvester-Gallai theorem states that for a finite set of points in the plane, if every line determined by any two of these points also contains a third, then the set is necessarily made of collinear points. In this paper, we first provide a counterexample in the plane when the point set is countably infinite but bounded. Then we consider a variant of the Sylvester-Gallai theorem where instead of a finite point set we have a finite family of convex sets in $\mathbb{R}^d$ ($d\geq 2$). Finally, we present another variant of the Sylvester-Gallai theorem, when instead of point sets we have a finite family of line-segments in the plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14518 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Excursions in Sylvester-Gallai land Barany, Imre Du, Julia Q. Schwarz, Dan Yuan, Liping Zamfirescu, Tudor Combinatorics 52C35, 51M04 The Sylvester-Gallai theorem states that for a finite set of points in the plane, if every line determined by any two of these points also contains a third, then the set is necessarily made of collinear points. In this paper, we first provide a counterexample in the plane when the point set is countably infinite but bounded. Then we consider a variant of the Sylvester-Gallai theorem where instead of a finite point set we have a finite family of convex sets in $\mathbb{R}^d$ ($d\geq 2$). Finally, we present another variant of the Sylvester-Gallai theorem, when instead of point sets we have a finite family of line-segments in the plane. |
| title | Excursions in Sylvester-Gallai land |
| topic | Combinatorics 52C35, 51M04 |
| url | https://arxiv.org/abs/2512.14518 |