An ordinary line of a point arrangement is a line containing exactly two of its points. Dirac (1951) conjectured that every sufficiently large
set of 
noncollinear points contains at least 
ordinary lines (Borwein and Bailey 2003, p. 18), and
Green and Tao (2013) proved this conjecture.
Csima and Sawyer (1993) proved that every arrangement of 
noncollinear points determines at least 
ordinary lines. The stronger all-
version of Dirac's bound has two known exceptions: the Kelly-Moser
configuration (7 points, 3 ordinary lines; cf. Fano plane)
and McKee's configuration (13 points, 6 ordinary lines).
More generally, a line spanned by a finite point set 
is 
-ordinary
if it contains at most 
points of 
,
and a 
-ordinary
triangle is a noncollinear triple whose three side lines are 
-ordinary. If 
, no line contains more than 
of the 
points, and the points cannot be covered by two lines, then
there are 
17-ordinary triangles (Dumitrescu and Pach 2026). They also gave an 
algorithm to count all 
-ordinary triangles when 
.
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).
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.Dumitrescu, A. and Pach, J. "Note on the Number of
Almost Ordinary Triangles." Elec. J. Combin. 33, No. 3, P3.33,
2026. 
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."