The Sylvester-Chvátal theorem
type: Item
instance of: scholarly article
title: The Sylvester-Chvátal theorem
zbMATH Open document ID: 1090.51005
author: Xiaomin Chen
published in: Discrete \& Computational Geometry
publication date: 2006-03-21T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 51F99
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 5013771
zbMATH Keywords: Sylvester theorem
zbMATH Keywords: lines in metric spaces
zbMATH Keywords: finite metric spaces
zbMATH Keywords: block design
reviewed by: Herbert Hotje
MaRDI profile type: Publication
full work available at URL: s00454-005-1216-9
OpenAlex ID: W1983961624
DOI: 10.1007/S00454-005-1216-9
description: scientific article; zbMATH DE number 5013771