On the finiteness of near polygons with 3 points on every line
type: Item
instance of: scholarly article
title: On the finiteness of near polygons with 3 points on every line
zbMATH Open document ID: 1034.51002
DOI: 10.1023/A:1025165325390
author: Bart De Bruyn
published in: Journal of Algebraic Combinatorics
publication date: 2003-09-25T00:00:00Z
review text: The author deals with near \(2d\)-gons. A {near \(2d\)-gon} is a partial linear space \(\Gamma = ({\mathcal P}, {\mathcal L}, I)\) satisfying the following two conditions: (1) For every point \(p \in {\mathcal P}\) and every line \(l \in {\mathcal L}\) there exists a unique point on \(l\) with minimal distance to \(p\), where the distance is measured in the collinearity graph of \(\Gamma\). (2) The diameter of \(\Gamma\) equals \(d\). A near \(2d\)-gon is called of order \((s, t)\), if each point is incident with exactly \(t+1\) lines and if each line is incident with exactly \(s+1\) points. \(s\) and \(t\) are not necessarily finite. Note that a near quadrangle is already a generalized quadrangle. The main result of de Bruyn is as follows: Theorem. Let \(\Gamma\) be a near \(2d\)-gon of order \((2, t)\) such that any two points at distance 2 have at least two common neighbours. Then \(\Gamma\) is finite.
reviewed by: Johannes Ueberberg
Mathematics Subject Classification ID: 51E12
zbMATH DE Number: 1985815
zbMATH Keywords: generalized quadrangles
zbMATH Keywords: generalized polygons
zbMATH Keywords: finiteness
zbMATH Keywords: near \(2d\)-gons
MaRDI profile type: Publication
cites work: Q3361478
cites work: Near polygons and Fischer spaces
cites work: The structure of near polygons with quads
cites work: On linear representations of near hexagons
cites work: Orbits of Permutation Groups on Unordered Sets, II
cites work: Q3344217
cites work: Near \(n\)-gons and line systems
cites work: Sur la trialité et certains groupes qui s'en déduisent
full work available at URL: a:1025165325390
OpenAlex ID: W12903319
description: scientific article; zbMATH DE number 1985815