Infinite linear spaces
type: Item
instance of: scholarly article
title: Infinite linear spaces
zbMATH Open document ID: 0803.51001
DOI: 10.1016/0012-365X(92)00503-J
author: Peter J. Cameron
published in: Discrete Mathematics
publication date: 1995-01-02T00:00:00Z
review text: While counting is the main tool in the theory of finite linear spaces in the infinite case free constructions and transfinite induction are used. Some well-known results are generalized in the paper under review: There exist free incidence structures of any infinite cardinality with the property that any \(t\) points lie in exactly \(\lambda\) blocks and any \(s>\lambda\) blocks intersect in exactly \(\mu<t\) points. The same holds for \(s \leq \lambda\), \(t \leq \mu\). Let \(X\) be an infinite linear space with \(\alpha\) lines in which each line has \(\alpha\) points. Let \(\mu\) be a cardinal number satisfying \(1<\mu < \alpha\). Then there exists \(S \subset X\) meeting every line in \(\mu\) points. Furthermore, among other things it is shown that there exists a countable infinite Steiner triple system \(S\) with 2-transitive automorphism group such that any finite triple system occurs as a subsystem of \(S\). Several problems are posed in connection with the theorems.
Mathematics Subject Classification ID: 51A05
Mathematics Subject Classification ID: 51B05
Mathematics Subject Classification ID: 51E10
Mathematics Subject Classification ID: 03E20
zbMATH DE Number: 599962
zbMATH Keywords: free extension
zbMATH Keywords: projective space
zbMATH Keywords: spread
zbMATH Keywords: infinite linear spaces
zbMATH Keywords: transfinite induction
zbMATH Keywords: Steiner triple system
reviewed by: Herbert Hotje
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 599962