On sets of sets mutually intersecting in exactly one element
type: Item
instance of: scholarly article
title: On sets of sets mutually intersecting in exactly one element
zbMATH Open document ID: 0966.51005
DOI: 10.1007/BF01220301
author: J. Eisfeld
published in: Journal of Geometry
publication date: 2001-08-06T00:00:00Z
review text: The author studies families \({\mathcal S}\) of subsets of some finite set \(M\) such that each \(S \in {\mathcal S}\) has bounded cardinality \(r\), any two distinct sets of \({\mathcal S}\) intersect in exactly one element and each \(m \in M\) is contained in at least two such subsets. The main result of the paper under review gives an upper bound \(b(r)\) for the cardinality of \({\mathcal S}\) as well as a characterization of such pairs \((M,{\mathcal S})\) that attain this upper bound. In the language of geometry this gives a characterization of finite linear spaces (almost) having the maximum number \(b(r)\) of lines. The author proves that \(b(r) = 2/27r^3 + 7/18 r^2 + 1/2 r + 1/27\). The proof of this result is based on double-counting and a rather complicated and detailed investigation of five different cases. If \({\mathcal L} = (M,{\mathcal S})\) is a linear space with \(|{\mathcal S }|= b(r)\), \(r \neq 4\), then \({\mathcal L}\) is a symmetrical tactical configuration with parameters \((r^2-r/2-1,r)\). If \({\mathcal L} = (M,{\mathcal S})\) is a linear space with \(|{\mathcal S }|= b(r) - 1/27\), \(r \neq 3\), then \({\mathcal L}\) is a symmetrical tactical configuration with parameters \((r^2 - (r+1)/2,r)\). For \(r=7\) an example of such a tactical configuration is known to exist, whereas very little is known about the \((r^2-r/2-1,r)\)-tactical configurations [see \textit{H. Gropp}, Ann. Discrete Math. 52, 227-239 (1992; Zbl 0767.05034)].
Mathematics Subject Classification ID: 51E20
zbMATH DE Number: 1443426
zbMATH Keywords: linear space
zbMATH Keywords: tactical configuration
MaRDI profile type: Publication
cites work: On sets of planes in projective spaces intersecting mutually in one point
cites work: On sets of \(n\)-dimensional subspaces of projective spaces intersecting mutually in an \((n-2)\)-dimensional subspace
cites work: Q4028434
full work available at URL: https://doi.org/10.1007/bf01220301
OpenAlex ID: W2117311692
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description: scientific article; zbMATH DE number 1443426