Regularity conditions on the transversals of lines in a partial geometry
type: Item
instance of: scholarly article
title: Regularity conditions on the transversals of lines in a partial geometry
zbMATH Open document ID: 0673.51012
DOI: 10.1016/S0195-6698(89)80046-0
author: Hans Gevaert
published in: European Journal of Combinatorics
publication date: 1989-01-01T00:00:00Z
review text: Let \(S=(P,B,I)\) be a partial geometry with parameters \(s, t, \alpha\) \((\alpha \ne t+1)\); that is, each line contains \(s+1\) points, through each point there are \(t+1\) lines and there are precisely \(\alpha\) lines joining any non-incident point and line. If \(S\) also satisfies the further property that, for any pair of skew lines \(\ell\) and \(m\) and any point \(P\) on neither \(\ell\) nor \(m\), there is a unique transversal of \(\ell\) and \m\) through \(P\), then \(S\) is one of three types: (1) a grid of order \(s+1\) with \(\alpha +1;\) (2) the points and lines of \(\mathrm{PG}(3,q)\) with \(\alpha =s+1;\) (3) the dual of a certain net of order 4 and degree 3 with \(\alpha =2\), \(s=2\), \(t=3\).
Mathematics Subject Classification ID: 51E30
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 4101982
zbMATH Keywords: partial geometry
zbMATH Keywords: transversals of lines
MaRDI profile type: Publication
cites work: A common characterization of finite projective spaces and affine planes
cites work: Strongly regular graphs, partial geometries and partially balanced designs
cites work: Q3888068
cites work: Finite linear spaces with a regularity condition on the transversals of two secant lines
cites work: Some characterizations of finite projective spaces
cites work: Q3344217
cites work: Q4154874
description: scientific article; zbMATH DE number 4101982