Excursions in Sylvester-Gallai land

TL;DR

This paper explores extensions and counterexamples related to the Sylvester-Gallai theorem, including infinite point sets, convex set variants, and line-segment configurations in the plane.

Contribution

It provides a counterexample for infinite bounded point sets and introduces new variants of the Sylvester-Gallai theorem involving convex sets and line-segments.

Findings

Counterexample for countably infinite bounded point sets in the plane.

New variants of Sylvester-Gallai theorem for convex sets in higher dimensions.

Results on line-segment configurations in the plane.

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.