Combinatorial properties of planar spaces and embeddability
type: Item
instance of: scholarly article
title: Combinatorial properties of planar spaces and embeddability
zbMATH Open document ID: 0605.51009
DOI: 10.1016/0097-3165(86)90068-3
author: Luc Teirlinck
published in: Journal of Combinatorial Theory. Series A
publication date: 1986-01-01T00:00:00Z
review text: Let S be a planar space i.e. a geometric lattice of rank 4. \textit{J. Kahn} [Math. Z, 175, 219-247 (1980; Zbl 0443.06013)] proved that if all points- residues of S are projective planes and if S satisfies a so called ''bundle axiom'', then S is embeddable in a 3-dimensional projective space. The present paper gives combinatorial conditions implying the embeddability of a planar space as a subset \({\mathcal P}\) of points of some (not necessarily 3-dimensional) projective space, together with the projective lines and planes spanned by points of \({\mathcal P}.\) Suppose each plane V of S is finite and contains \(N(V)+1\) lines through each point, where N(V) depends only of the choice of V. Suppose further that, for every plane V, all lines of V contain at least \((2N(V)+5)/3\) points. Then, the author proofs that N(V) does not depend on V and that S is locally projective and embeddable in a projective space. The proof uses other rather technical conditions implying embeddabiliy for a planar space. Let us mention that Buekenhout's characterization of affine spaces of order \(k\geq 4\) as linear spaces whose planes are affine is a corollary of the main result of the present paper whenever \(k\geq 5\).
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 51D20
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 3977824
zbMATH Keywords: embeddability of planar spaces in projective spaces
zbMATH Keywords: combinatorial conditions
reviewed by: Christiane Lefèvre-Percsy
MaRDI profile type: Publication
full work available at URL: 0097-3165(86)90068-3
OpenAlex ID: W2031463330
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description: scientific article; zbMATH DE number 3977824