Discrete & Computational Geometry

, Volume 50, Issue 2, pp 409–468

On Sets Defining Few Ordinary Lines

Article

DOI: 10.1007/s00454-013-9518-9

Cite this article as:
Green, B. & Tao, T. Discrete Comput Geom (2013) 50: 409. doi:10.1007/s00454-013-9518-9

Abstract

Let \(P\) be a set of \(n\) points in the plane, not all on a line. We show that if \(n\) is large then there are at least \(n/2\)ordinary lines, that is to say lines passing through exactly two points of \(P\). This confirms, for large \(n\), a conjecture of Dirac and Motzkin. In fact we describe the exact extremisers for this problem, as well as all sets having fewer than \(n-C\) ordinary lines for some absolute constant \(C\). We also solve, for large \(n\), the “orchard-planting problem”, which asks for the maximum number of lines through exactly 3 points of \(P\). Underlying these results is a structure theorem which states that if \(P\) has at most \(Kn\) ordinary lines then all but O(K) points of \(P\) lie on a cubic curve, if \(n\) is sufficiently large depending on \(K\).

Keywords

Sylvester–Gallai Ordinary lines Cubic curves Dirac–Motzkin 

Mathematics Subject Classification (1991)

20G40 20N99 

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Centre for Mathematical SciencesCambridgeEngland, UK
  2. 2.Department of MathematicsUCLALos AngelesUSA

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