Ordinary Line

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 .

da Silva and Fukuda conjectured that every noncollinear, line-separable two-color arrangement with color classes whose sizes differ by at most one has a bichromatic ordinary line. Finschi and Fukuda (2001) found a unique nine-point counterexample, up to isomorphism, among 15296266 distinct configurations. Its color classes have four and five points; the equal-size case is not refuted by this example.

For coordinates in the complex numbers, a nonconcentration hypothesis gives a quadratic lower bound. Kroeker and Lund (2026) proved that distinct points in determine at least ordinary lines if no affine plane contains more than of the points. The constant is not known to be sharp.