On multi-coloured lines in a linear space
type: Item
instance of: scholarly article
title: On multi-coloured lines in a linear space
zbMATH Open document ID: 0616.05003
DOI: 10.1016/0097-3165(85)90053-6
published in: Journal of Combinatorial Theory. Series A
publication date: 1985-01-01T00:00:00Z
review text: A linear space (P,L) consists of a set of points P and a set of lines L; a line is a subset of the points. Furthermore, every distinct pair of points belongs to exactly one line in L. (Note that distinct lines intersect in at most one point.) Suppose \(X_ 1,...,X_ k\subset P\) and let \(L(X_ 1,...,X_ k)\) be the set of lines which have points in at least two on the X's. The author proves that if \(X_ 1,...,X_ k\subseteq P\) are distinct m-sets and \(X_ 1\cup...\cup X_ k\) is not contained in a line, then \(| L(X_ 1,...,X_ k)| \geq (k-1)n\), and a characterization of when equality holds is also given. A note added in proof asserts that Z. Füredi and I. G. Rosenberg have proved the author's conjectured generalization: Suppose \(X_ 1,...,X_ 2\) are disjoint sets of points with \(| X_ 1| \geq...\geq | X_ 2|\) and \(X_ 1\cup...\cup X_ k\) is not contained in a line, then \(| L(X_ 1,...,X_ k)| \geq | X_ 2| +...+| X_ k|\) and equality only occurs in the way already described in the author's theorem.
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 51A05
zbMATH DE Number: 3997797
zbMATH Keywords: linear space
author: Roy Meshulam
reviewed by: David A. Klarner
MaRDI profile type: Publication
full work available at URL: 0097-3165(85)90053-6
OpenAlex ID: W2069510826
cites work: Q3880849
description: scientific article; zbMATH DE number 3997797