 Szemerédi–Trotter theorem

For $m$ lines and $n$ points in the real plane, the number of point-line incidences isConnecting consecutive incidence points on each line and applying the Crossing lemma proves the bound.

The Szemerédi–Trotter theorem is a fundamental result in combinatorial geometry that provides bounds on the incidences between points and lines in the plane. Specifically, it addresses how many points lie on a set of lines, providing a relationship between three parameters: the number of points, the number of lines, and the number of incidences (that is, points that lie on those lines).