On multi-coloured lines in a linear space
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.
scientific article; zbMATH DE number 3997797