Planar and affine spaces


ANAPA SABAN P., GÜNALTILI İ., Van Maldeghem H.

DISCRETE MATHEMATICS, vol.309, no.9, pp.2897-2904, 2009 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 309 Issue: 9
  • Publication Date: 2009
  • Doi Number: 10.1016/j.disc.2008.07.024
  • Journal Name: DISCRETE MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.2897-2904
  • Eskisehir Osmangazi University Affiliated: Yes

Abstract

In this note, we characterize finite three-dimensional affine spaces as the only linear spaces endowed with set Omega of proper subspaces having the properties (1) every line contains a constant number of points, say n, with n > 2: (2) every triple of noncollinear points is contained in a unique member of Omega; (3) disjoint or coincide is an equivalence relation in Omega with the additional property that every equivalence class covers all points. We also take a look at the case n = 2 (in which case we have a complete graph endowed with a set Q of proper complete subgraphs) and classify these objects: besides the affine 3-space of order 2, two small additional examples turn up. Furthermore, we generalize our result in the case of dimension greater than three to obtain a characterization of all finite affine spaces of dimension at least 3 with lines of size at least 3. (C) 2008 Elsevier B.V. All rights reserved.