De Bruijn–Erdős theorem (incidence geometry)


In incidence geometry, the De Bruijn–Erdős theorem, originally published by, 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.
Although the proof given by De Bruijn and Erdős is combinatorial, De Bruijn and Erdős noted in their paper that the analogous result is a consequence of the Sylvester–Gallai theorem, by an induction on the number of points.

Statement of the theorem

Let P be a configuration of n points in a projective plane, not all on a line. Let t be the number of lines determined by P. Then,
The theorem is clearly true for three non-collinear points. We proceed by induction.
Assume n > 3 and the theorem is true for n − 1.
Let P be a set of n points not all collinear.
The Sylvester–Gallai theorem states that there is a line containing exactly two points of P. Such two point lines are called ordinary lines.
Let a and b be the two points of P on an ordinary line.
If the removal of point a produces a set of collinear points then P generates a near pencil of n lines.
Otherwise, the removal of a produces a set, P' , of n − 1 points that are not all collinear.
By the induction hypothesis, P' determines at least n − 1 lines. The ordinary line determined by a and b is not among these, so P determines at least n lines.

J. H. Conway's proof

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