Dual blocking sets in projective and affine planes
type: Item
instance of: scholarly article
title: Dual blocking sets in projective and affine planes
zbMATH Open document ID: 0659.51013
DOI: 10.1007/BF00151351
author: Francesco Mazzocca
author: Roy Meshulam
author: Peter J. Cameron
published in: Geometriae Dedicata
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathcal F}\) be the set of lines of a projective or affine plane. A blocking set for \({\mathcal F}\) is a set which meets every member of \({\mathcal F}\) but contains none. A dual blocking set for \({\mathcal F}\) is a set which meets every blocking set for \({\mathcal F}\) but contains no member of \({\mathcal F}\). A line oval in a projective plane of order n is a set of \(n+2\) lines, no three concurrent. A line oval in an affine plane of order n is a line oval in the corresponding projective plane, one of whose lines is the line at infinity; in other words, a set of \(n+1\) lines, no three concurrent and no two parallel. The authors prove the following theorems. Theorem 1. Let S be a dual blocking set in a projective plane of order \(n\geq 3\). Then \(| S| \leq n(n+1)/2\). Equality holds if and only if either (a) \(S=\cup {\mathcal L}\setminus L\), where \({\mathcal L}\) is a line oval and \(L\in {\mathcal L}\); or (b \(n=3\) and S is the complement of the union of two lines. Theorem 2. Let S be a dual blocking set in an affine plane of order \(n\geq 4\). Then \(| S| \leq n(n-1)/2\). Equality holds if and only if \(S=\cup {\mathcal L}\setminus L\), where \({\mathcal L}\) is a line oval and \(L\in {\mathcal L}\).
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 4078133
zbMATH Keywords: projective and affine planes. dual blocking set
zbMATH Keywords: line oval
reviewed by: Andreiy Kondrat'yev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00151351
OpenAlex ID: W1986624173
description: scientific article; zbMATH DE number 4078133