Discrete & Computational Geometry

, Volume 50, Issue 2, pp 409–468 | Cite as

On Sets Defining Few Ordinary Lines

Article

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 

References

  1. 1.
    Bacharach, I.: Ueber den Cayley’schen Schnittpunktsatz. Math. Ann. 26, 275–299 (1886)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Beck, J.: On the lattice property of the plane and some problems of Dirac, Motzkin, and Erdos in combinatorial geometry. Combinatorica 3, 281–297 (1983)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bix, R.: Conics and Cubics: A Concrete Introduction to Algebraic Curves. Undergraduate Texts in Mathematics, 2nd edn. Springer, New York (2006)Google Scholar
  4. 4.
    Borwein, P., Moser, W.O.J.: A survey of Sylvester’s problem and its generalizations. Aequationes Math. 40(2–3), 111–135 (1990)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Burr, S., Grünbaum, B., Sloane, N.J.: The orchard problem. Geom. Dedicata 2, 397–424 (1974)MATHCrossRefGoogle Scholar
  6. 6.
    Carnicer, J.M., Gasca, M.: Cubic pencils of lines and bivariate interpolation. J. Comput. Appl. Math. 219(2), 370–382 (2008)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Carnicer, J.M., Godés, C.: Generalised principal lattices and cubic pencils. Num. Algorithms 44(2), 133–145 (2007)MATHCrossRefGoogle Scholar
  8. 8.
    Cayley, A.: On the Intersection of Curves. Cambridge University Press, Cambridge (1889)Google Scholar
  9. 9.
    Chasles, M.: Traité des Sections Coniques. Gauthier-Villars, Paris (1885)Google Scholar
  10. 10.
    Crowe, D.W., McKee, T.A.: Sylvester’s problem on collinear points. Math. Mag. 41, 30–34 (1968)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Csima, J., Sawyer, E.T.: There exist \(6n/13\) ordinary points. Discrete Comput. Geom. 9(2), 187–202 (1993)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Dirac, G.: Collinearity properties of sets of points. Q. J. Math. 2, 221–227 (1951)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Eisenbud, D., Green, M., Harris, J.: Cayley–Bacharach theorems and conjectures. Bull. Am. Math. Soc. 33, 295–324 (1996)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Elekes, G., Nathanson, M., Ruzsa, I.: Convexity and sumsets. J. Number Theory 83(2), 194–201 (2000)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Erdős, P., Purdy, G.: Extremal problems in combinatorial geometry. In: Handbook of Combinatorics, vol. 1, pp. 809–874. Elsevier, Amsterdam (1995)Google Scholar
  16. 16.
    Fournier, J.J.F.: Sharpness in Young’s inequality for convolution. Pac. J. Math. 72(2), 383–397 (1977)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Freĭman, G.: Foundations of a structural theory of set addition. Translated from the Russian. Translations of Mathematical Monographs, vol. 37, vii+108 pp. American Mathematical Society, Providence, RI (1973)Google Scholar
  18. 18.
    Gallai, T.: Solution to problem number 4065. Am. Math. Mon. 51, 169–171 (1944)CrossRefGoogle Scholar
  19. 19.
    Gowers, W.T.: A new proof of Szemerédi’s theorem for arithmetic progressions of length four. Geom. Funct. Anal. 8(3), 529–551 (1998)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Hansen, S.: Contributions to the Sylvester–Gallai theory. Ph.D. Thesis, University of Copenhagen, Copenhagen (1981)Google Scholar
  21. 21.
    Jackson, J.: Rational Amusement for Winter Evenings. Longman, Hurst, Rees, Orme and Brown, London (1821)Google Scholar
  22. 22.
    Kelly, L., Moser, W.: On the number of ordinary lines determined by \(n\) points. Can. J. Math. 10, 210–219 (1958)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Kneser, M.: Abschätzung der asymptotischen Dichte von Summenmengen. Math. Zeit. 58, 459–484 (1953)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Kneser, M.: Ein Satz über abelschen Gruppen mit Anwendungen auf die Geometrie der Zahlen. Math. Zeit. 61, 429–434 (1955)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Melchior, E.: Über Vielseite der projektiven Ebene. Deutsche Math. 5, 461–475 (1940)MathSciNetGoogle Scholar
  26. 26.
    Motzkin, Th: The lines and planes connecting the points of a finite set. Trans. Am. Math. Soc. 70, 451–464 (1951)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    Nilakantan, N.: Extremal problems related to the Sylvester–Gallai theorem. In: Combinatorial and Computational Geometry. MSRI Publications, Berkeley (2005)Google Scholar
  28. 28.
    Pach, J., Sharir, M.: Combinatorial geometry and its algorithmic applications: the Alcalá lectures. In: AMS Mathematical Surveys and Monographs, vol. 152, 235 pp. American Mathematical Society, Providence, RI (2009)Google Scholar
  29. 29.
    Poonen, B., Rubinstein, M.: The number of intersection points made by the diagonals of a regular polygon. SIAM J. Discrete Math. 11(1), 135–156 (1998)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Silverman, J., Tate, J.: Rational Points on Elliptic Curves. Undergraduate Texts in Mathematics, \(\text{ x }+281\) pp. Springer, New York (1992) Google Scholar
  31. 31.
    Sudakov, B., Szemerédi, E., Vu, V.: On a problem of Erdős and Moser. Duke Math. J. 129(1), 129–154 (2005)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Sylvester, J.: Mathematical Question 2571. Educational Times (1868)Google Scholar
  33. 33.
    Sylvester, J.: Mathematical Question 11851. Educational Times (1893)Google Scholar
  34. 34.
    Tao, T.C.: Pappus’s theorem and elliptic curves. http://terrytao.wordpress.com/2011/07/15/pappuss-theorem-and-elliptic-curves/
  35. 35.
    Tao, T.C., Vu, V.H.: Additive combinatorics. In: Cambridge Studies in Advanced Mathematics, vol. 105, xviii+512 pp. Cambridge University Press, Cambridge (2006)Google Scholar

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

Personalised recommendations