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arXiv · 2512.14518

Excursions in Sylvester-Gallai land

Imre Barany·Julia Q. Du·Dan Schwarz·Liping Yuan·Tudor Zamfirescu
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Abstract

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.

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Imre Barany, Julia Q. Du, Dan Schwarz, Liping Yuan, Tudor Zamfirescu. 2025-12-16. Excursions in Sylvester-Gallai land. https://arxiv.org/abs/2512.14518

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