Discrete & Computational Geometry

, Volume 9, Issue 2, pp 187–202 | Cite as

There exist 6n/13 ordinary points

  • J. Csima
  • E. T. Sawyer


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.


Additional Line Discrete Comput Geom Euclidean Plane Open Segment Ordinary Point 
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  1. [BM]
    P. Borwein and W. O. J. Moser, A survey of Sylvester's problem and its generalizations,Aequationes Math. 40, 111–135 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  2. [C1]
    H. S. M. Coxeter, A problem of collinear points,Amer. Math. Monthly 55, 26–28 (1948).MR 9, p. 458.MathSciNetCrossRefGoogle Scholar
  3. [C2]
    H. S. M. Coxeter,Introduction to Geometry, 2nd edn., Wiley, New York, 1969.zbMATHGoogle Scholar
  4. [CM]
    D. W. Crowe and T. A. McKee, Sylvester's problem on collinear points,Math. Mag. 41, 30–34 (1968).MR 38, #3761.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [D1]
    G. A. Dirac, Collinearity properties of sets of points,Quart. J. Math. 2, 221–227 (1951).MR 13, p. 270.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [D2]
    G. A. Dirac, Review of Kelly and Moser (1958),MR 20, #3494 (1959).Google Scholar
  7. [E1]
    P. Erdös, Problems for Solution, #4065,Amer. Math. Monthly 50, 65 (1943).MathSciNetCrossRefGoogle Scholar
  8. [E2]
    P. Erdös, Solution of Problem 4065,Amer. Math. Monthly 51, 169–171 (1944).MathSciNetCrossRefGoogle Scholar
  9. [E3]
    P. Erdös, Personal reminiscences and remarks on the mathematical work of Tibor Gallai,Combinatorica 2, 207–212 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  10. [EP]
    P. Erdös and G. Purdy, Some extremal problems in combinatorial geometry, Preprint distributed at the 20th Southeastern Conference on Combinatorics, Graph Theory, and Computing, held at Florida Atlantic University, Feb. 20–24, 1989.Google Scholar
  11. [G]
    B. Grünbaum, The importance of being straight,Proc. 12th Internat. Sem. Canad. Math. Congress, Vancouver, 1969.Google Scholar
  12. [H]
    S. Hansen, Contributions to the Sylvester-Gallai theory, Dissertation for the habilitation, University of Copenhagen, 100 copies privately printed, 1981.Google Scholar
  13. [K]
    A. B. Kempe, On the geographical problem of the four colours,Amer. J. Math. 2, 193–200 (1879).MathSciNetCrossRefGoogle Scholar
  14. [KM]
    L. M. Kelly and W. O. J. Moser, On the number of ordinary lines determined byn points,Canad. J. Math. 10, 210–219 (1958).MR 20, #3494.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [KR]
    L. M. Kelly and R. R. Rottenberg, Simple points on pseudoline arrangements,Pacific J. Math. 40, 617–622 (1972).MR 46, #6150.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [L]
    G. D. W. Lang, The dual of a well-known theorem,Math. Gaz. 39, 314 (1955).CrossRefGoogle Scholar
  17. [Me]
    E. Melchior, Uber Vielseite der projektiven Ebene,Deutsche Math. 5, 461–475 (1940).MR 3, p. 13.MathSciNetzbMATHGoogle Scholar
  18. [Mo]
    W. O. J. Moser, Abstract groups and geometrical configurations, Ph.D. thesis, University of Toronto, 1957.Google Scholar
  19. [Mot]
    Th. Motzkin, The lines and planes connecting the points of a finite set,Trans. Amer. Math. Soc. 70, 451–464 (1951).MR 12, p. 849.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [St]
    R. Steinberg, Three point collinearity,Amer. Math. Monthly 51, 169–171 (1944).MathSciNetCrossRefGoogle Scholar
  21. [Sy]
    J. J. Sylvester, Mathematical Question 11851,Educational Times, Vol. 46, March, p. 156 (1893).Google Scholar
  22. [VY]
    O. Veblen and J. W. Young,Projective Geometry, Vol. 1, Ginn, Boston, 1910.zbMATHGoogle Scholar
  23. [W1]
    H. J. Woodall, Solution to Question #11851,Educational Times, Vol. 46, May, p. 231 (1893).Google Scholar
  24. [W2]
    H. J. Woodall, Solution to Question #11851,Mathematical Questions and Solutions, from the “Educational Times”, edited by W. J. C. Miller, Vol. 59, p. 98, Hodgson, London, 1893.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • J. Csima
    • 1
  • E. T. Sawyer
    • 1
  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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