Linear Spaces with Few Lines
A famous theorem in the theory of linear spaces states that every finite linear space has at least as many lines as points. This result of De Bruijn and Erd|s led to the conjecture that every linear space with "few lines" canbe obtained from a projective
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen
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Klaus Metsch
Linear Spaces with Few Lines
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Klaus Metsch Mathematisches Institut Justus-Liebig-Universitat ArndtstraBe 2, W-6300 GieBen, FRG
Mathematics Subject Classification (1991): 51E20
ISBN 3-540-54720-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54720-7 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Typesetting: Camera ready by author Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 46/3140-543210 Printed on acid-free paper
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INTRODUCTION
It is known since 40 years that a linear space has at least as many lines as
points with equality only if it is a generalized projective plane. This result of de Bruijn and Erdos (1948) led to the conjecture that every linear space with "few lines" can be obtained from a certain projective plane by changing only a small part of its structure. It is
surprising that it took more than
Bridges (1972) showed that every linear space with b = v+ 1
'*
20 years until
6 (b the number of
lines, v the number of points) is a punctured projective plane. However, since then many results have been obtained. It is the main purpose of this paper to study systematically this embedding problem. In particular, we shall collect the old results and present quite a few new ones. We shall, however, also study linear spaces with few lines which have no natural embedding in a projective plane.
When studying finite linear spaces which have a chance to be embeddable in a projective plane of order n, it is sensible to suppose that b the number of lines in a projective plane of order n)
and v
n 2+n+ 1 (which is (n -I)2+(n -1)+2
= n 2-n+2 (this is due to the fa
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