There exist 6n/13 ordinary points
 J. Csima,
 E. T. Sawyer
 … show all 2 hide
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Abstract
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 KellyMoser example (a complete quadrilateral with its three diagonal lines). In 1981 S. Hansen claimed to have improved this ton/2 (apart from pencils, the KellyMoser 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 KellyMoser example), with equality in the McKee example. Our result applies also to arrangements of pseudolines.
 P. Borwein and W. O. J. Moser, A survey of Sylvester's problem and its generalizations,Aequationes Math. 40, 111–135 (1990).
 H. S. M. Coxeter, A problem of collinear points,Amer. Math. Monthly 55, 26–28 (1948).MR 9, p. 458.
 H. S. M. Coxeter,Introduction to Geometry, 2nd edn., Wiley, New York, 1969.
 D. W. Crowe and T. A. McKee, Sylvester's problem on collinear points,Math. Mag. 41, 30–34 (1968).MR 38, #3761.
 G. A. Dirac, Collinearity properties of sets of points,Quart. J. Math. 2, 221–227 (1951).MR 13, p. 270.
 G. A. Dirac, Review of Kelly and Moser (1958),MR 20, #3494 (1959).
 P. Erdös, Problems for Solution, #4065,Amer. Math. Monthly 50, 65 (1943).
 P. Erdös, Solution of Problem 4065,Amer. Math. Monthly 51, 169–171 (1944).
 P. Erdös, Personal reminiscences and remarks on the mathematical work of Tibor Gallai,Combinatorica 2, 207–212 (1982).
 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.
 B. Grünbaum, The importance of being straight,Proc. 12th Internat. Sem. Canad. Math. Congress, Vancouver, 1969.
 S. Hansen, Contributions to the SylvesterGallai theory, Dissertation for the habilitation, University of Copenhagen, 100 copies privately printed, 1981.
 A. B. Kempe, On the geographical problem of the four colours,Amer. J. Math. 2, 193–200 (1879).
 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.
 L. M. Kelly and R. R. Rottenberg, Simple points on pseudoline arrangements,Pacific J. Math. 40, 617–622 (1972).MR 46, #6150.
 G. D. W. Lang, The dual of a wellknown theorem,Math. Gaz. 39, 314 (1955).
 E. Melchior, Uber Vielseite der projektiven Ebene,Deutsche Math. 5, 461–475 (1940).MR 3, p. 13.
 W. O. J. Moser, Abstract groups and geometrical configurations, Ph.D. thesis, University of Toronto, 1957.
 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.
 R. Steinberg, Three point collinearity,Amer. Math. Monthly 51, 169–171 (1944).
 J. J. Sylvester, Mathematical Question 11851,Educational Times, Vol. 46, March, p. 156 (1893).
 O. Veblen and J. W. Young,Projective Geometry, Vol. 1, Ginn, Boston, 1910.
 H. J. Woodall, Solution to Question #11851,Educational Times, Vol. 46, May, p. 231 (1893).
 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.
 Title
 There exist 6n/13 ordinary points
 Journal

Discrete & Computational Geometry
Volume 9, Issue 1 , pp 187202
 Cover Date
 19931201
 DOI
 10.1007/BF02189318
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 J. Csima ^{(1)}
 E. T. Sawyer ^{(1)}
 Author Affiliations

 1. Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada