A Helly type theorem for abstract projective geometries
type: Item
instance of: scholarly article
title: A Helly type theorem for abstract projective geometries
zbMATH Open document ID: 1211.52007
author: Javier Bracho
author: Jorge L. Arocha
published in: Discrete \& Computational Geometry
publication date: 2011-03-10T00:00:00Z
review text: A flat in an abstract (axiomatic) projective geometry is a point set \(f\) such that any line \(\ell\) that shares two points with \(f\) is already contained in \(f\). The rank of \(f\) is the minimal cardinality of a point set \(A\) such that \(f\) equals the intersection of all flats containing \(A\). This rank is also defined for the projective geometry itself. The authors prove the following result for projective geometries of rank \(n\): ``A family of linear partitions has non-empty intersection if every \(\lfloor \frac{3n}{2} \rfloor\) members of the family have non-empty intersection.'' Here, a linear partition is the union of a family of flats whose span (the intersection of all flats containing all elements of their union) is of maximal rank. This extends a result by \textit{J. L. Arocha, J. Bracho} and \textit{L. Montejano} [Adv. Math. 213, No.~2, 902--918 (2007; Zbl 1121.52016)] where the same statement was shown to hold true for projective geometries over a (commutative) field.
reviewed by: Hans-Peter Schröcker
Mathematics Subject Classification ID: 52A35
Mathematics Subject Classification ID: 51A05
zbMATH DE Number: 5864110
zbMATH Keywords: Helly type theorem
zbMATH Keywords: projective geometry
zbMATH Keywords: linear partition
zbMATH Keywords: modular lattice
MaRDI profile type: Publication
full work available at URL: s00454-010-9287-7
OpenAlex ID: W1989956876
cites work: Flat transversals to flats and convex sets of a fixed dimension
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cites work: On the Foundations of Combinatorial Theory II. Combinatorial Geometries
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cites work: Connectivity in Matroids
DOI: 10.1007/S00454-010-9287-7
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description: scientific article; zbMATH DE number 5864110