Finite linear spaces and projective planes

Erdős, Pál and Mullin, R. C. and T. Sós, Vera and Stinson, D. R. (1983) Finite linear spaces and projective planes. DISCRETE MATHEMATICS, 47 (1). pp. 49-62. ISSN 0012-365X

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Abstract

In 1948, De Bruijn and Erdös proved that a finite linear space on ν points has at least ν lines, with equality occurring if and only if the space is either a near-pencil (all points but one collinear) or a projective plane. In this paper, we study finite linear spaces which are not near-pencils. We obtain a lower bound for the number of lines (as a function of the number of points) for such linear spaces. A finite linear space which meets this bound can be obtained provided a suitable projective plane exists. We then investigate the converse: can a finite linear space meeting the bound be embedded in a projective plane. © 1983.

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