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).
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. Monthly55, 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.
Monthly51, 169, 1944.Felsner, S. Geometric
Graphs and Arrangements. Wiesbaden, Germany: Vieweg, 2004.Guy,
R. K. "Unsolved Problems Come of Age." Amer. Math. Monthly96,
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 Times59, 98,
1893.
noncollinear points contains at least 
ordinary lines (Borwein and Bailey 2003, p. 18).
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).
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. 
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."