Ordinary Line (original) (raw)
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Given an arrangement of points, a line containing just two of them is called an ordinary line. Dirac (1951) conjectured that every sufficiently set of 
noncollinear points contains at least 
ordinary lines (Borwein and Bailey 2003, p. 18).
Csima and Sawyer (1993) proved that for an 
arrangement of points, at least 
lines must be ordinary. Only two exceptions are known for Dirac's conjecture: the Kelly-Moser configuration (7 points, 3 ordinary lines; cf. Fano plane) and McKee's configuration (13 points, 6 ordinary lines).
Silva and Fukuda conjectured that for any noncollinear, equally distributed, line-separable arrangement of points of two colors, there is at least one bichromatic ordinary line. Finschi and Fukuda found a unique nine-point counterexample in a study of 15296266 distinct configurations (Malkevitch).
See also
Collinear, General Position, Incident, Near-Pencil,Ordinary Line Graph, Ordinary Point, Special Point, Sylvester Graph
Portions of this entry contributed by Herve Bronnimann
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References
Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Coxeter, H. S. M. "A Problem of Collinear Points." Amer. Math. Monthly 55, 26-28, 1948.Coxeter, H. S. M. The Real Projective Plane, 3rd ed. Cambridge, England: Cambridge University Press, 1993.Csima, J. and Sawyer, E. "There Exist 
Ordinary Points." Disc. Comput. Geom. 9, 187-202, 1993.de Bruijn, N. G. and Erdős, P. "On a Combinatorial Problem." Hederl. Adad. Wetenach. 51, 1277-1279, 1948.Dirac, G. A. "Collinearity Properties of Sets of Points." Quart. J. Math. 2, 221-227, 1951.Erdős, P. "Problem 4065." Amer. Math. Monthly 51, 169, 1944.Felsner, S. Geometric Graphs and Arrangements. Wiesbaden, Germany: Vieweg, 2004.Guy, R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly 96, 903-909, 1989.Kaneko, A. and Kano, M. "Discrete Geometry on Red and Blue Points in the Plane." http://gorogoro.cis.ibaraki.ac.jp/web/papers/kano2003-48.pdf.Kelly, L. M. and Moser, W. O. J. "On the Number of Ordinary Lines Determined by 
Points." Canad. J. Math. 1, 210-219, 1958.Lang, D. W. "The Dual of a Well-Known Theorem."Math. Gaz. 39, 314, 1955.Malkevitch, J. "A Discrete Geometrical Gem." http://www.ams.org/featurecolumn/archive/sylvester1.html.Motzkin, T. "The Lines and Planes Connecting the Points of a Finite Set." Trans. Amer. Math. Soc. 70, 451-463, 1951.Sylvester, J. J. "Mathematical Question 11851." Educational Times 59, 98, 1893.
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Bronnimann, Herve and Weisstein, Eric W. "Ordinary Line." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/OrdinaryLine.html