Linear spaces and partitioning the projective plane
type: Item
instance of: scholarly article
title: Linear spaces and partitioning the projective plane
zbMATH Open document ID: 0885.51009
author: Ferenc Fodor
published in: Journal of Combinatorial Theory. Series A
publication date: 1997-12-15T00:00:00Z
review text: Let \({\mathcal P}=(P,\mathcal L)\) be a finite projective plane of order \(n\). Let \(Q\subset P\) be of cardinality at most \(2n-1\) and let \(\Pi\subset\mathcal L\) with \(|\Pi|<n+1+\sqrt n\). \textit{K. Metsch} [Lect. Notes Math. 1490, Springer, Berlin (1991; Zbl 0744.51005)] proved that the lines of \(\Pi\) are concurrent, if no point row (i.e. all points incident with some line) of \(\Pi\) is contained in \(Q\) and every point outside \(Q\) lies on exactly one line of \(\Pi\). It is easy to see that the bound on \(|Q|\) is almost sharp by considering a plane \(\mathcal P\) which possesses a Baer subplane, i.e. a subplane of order \(\sqrt n\). This gives \(Q\) with \(|Q|=2n+1\). The author proves that in the case \(|Q|=2n\) Metsch's result is still valid with the exception of one single configuration.
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 52C10
zbMATH DE Number: 1076900
zbMATH Keywords: linear space
zbMATH Keywords: projective plane
zbMATH Keywords: partitioning
MaRDI profile type: Publication
OpenAlex ID: W1974211667
cites work: Linear spaces with few lines
cites work: Finite linear spaces and projective planes
cites work: Q4163237
DOI: 10.1006/JCTA.1997.2772
description: scientific article; zbMATH DE number 1076900