Zbl.No: 521.51005
Autor: Erdös, Paul; Mullin, R.C.; Sos, V.T.; Stinson, D.R.
Title: Finite linear spaces and projective planes. (In English)
Source: Discrete Math. 47, 49-62 (1983).
Review: A nondegenerate finite linear space (NLS) is defined such that every two points occur in a unique line, every line contains \geq 2 points, and no line contains all but one of the points. The number of lines and of points are, respectively, called b and v. N.G.de Bruijn and P.Erdös [Indag. Math. 10, 1277-1279 (1948; Zbl 032.24405)] have shown that b \geq v holds in an NLS, with equality iff the NLS is a projective plane. In the paper under review, the authors show that in an NLS with v \geq 5 one has b \geq B(v) where
B(v) =
n2+n+1 if n2+2 \leq v \leq n2+n+1
n2+n if n2-n+3 \leq v \leq n2+1
n2+n-1 if v = n2-n+2,
with equality if n is the order of a projective plane. Moreover, minimal NLS's (that is, if no NLS on v points has fewer lines) and their embeddability in projective planes are studied. The paper concludes with a few open problems.
Reviewer: R.Artzy
Classif.: * 51A45 Incidence structures imbeddable into projective geometries 05B25 Finite geometries (combinatorics) 51E15 Affine and projective planes
Keywords: linear space; point; line; projective plane
Citations: Zbl.032.244
