Discrete & Computational Geometry

, Volume 9, Issue 2, pp 187-202

First online:

There exist 6n/13 ordinary points

  • J. CsimaAffiliated withDepartment of Mathematics and Statistics, McMaster University
  • , E. T. SawyerAffiliated withDepartment of Mathematics and Statistics, McMaster University

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In 1958 L. M. Kelly and W. O. J. Moser showed that apart from a pencil, any configuration ofn lines in the real projective plane has at least 3n/7 ordinary or simple points of intersection, with equality in the Kelly-Moser example (a complete quadrilateral with its three diagonal lines). In 1981 S. Hansen claimed to have improved this ton/2 (apart from pencils, the Kelly-Moser example and the McKee example). In this paper we show that one of the main theorems used by Hansen is false, thus leavingn/2 open, and we improve the 3n/7 estimate to 6n/13 (apart from pencils and the Kelly-Moser example), with equality in the McKee example. Our result applies also to arrangements of pseudolines.