De Bruijn–Erdős theorem (incidence geometry)

In incidence geometry, the De Bruijn–Erdős theorem, originally published by Nicolaas Govert de Bruijn and Paul Erdős in 1948,[1] states a lower bound on the number of lines determined by n points in a projective plane.

By duality, this is also a bound on the number of intersection points determined by a configuration of lines.

[2] Although the proof given by De Bruijn and Erdős is combinatorial, De Bruijn and Erdős noted in their paper that the analogous (Euclidean) result is a consequence of the Sylvester–Gallai theorem, by an induction on the number of points.

Let t be the number of lines determined by P. Then, The theorem is clearly true for three non-collinear points.

John Horton Conway has a purely combinatorial proof which consequently also holds for points and lines over the complex numbers, quaternions and octonions.