The Sylvester-Chvátal theorem
The Sylvester-Gallai theorem asserts that every finite set \(S\) of points in two-dimensional Euclidean space includes two points, \(a\) and \(b\), such that either there is no other point in \(S\) on the line \(ab\), or the line \(ab\) contains all the points in \(S\). Chvátal extended the notion of lines to arbitrary metric spaces and made a conjecture that generalizes the Sylvester-Gallai theorem. While the theorem is no longer true in arbitrary finite spaces in which lines are defined in the traditional (Menger) way, with \textit{V. Chvátal}'s recursive definition of lines [Discrete Comput. Geom. 31, No. 2, 175--195 (2004; Zbl 1053.51006)] the theorem can be proved. Thus no \((v,k,1)\) design as for example a finite affine or projective plane of order \(>2\) or a Steiner triple system is realizable as a metric space.
scientific article; zbMATH DE number 5013771