Beck's theorem

Remarks

The proof counts pairs of points of $P$ according to the number of points on the line through them. The Szemerédi–Trotter theorem bounds the number of lines containing between $k$ and $2k$ points of $P$ by $O(n^2/k^3 + n/k)$. Sum this bound over dyadic ranges of $k$. Unless some line is very rich, a constant fraction of the $\binom{n}{2}$ pairs then lie on lines with few points. Those lines are then of order $n^2$ in number. Example: if $P$ consists of $n - 1$ collinear points and one point off their line, clause 1 holds, while $P$ determines only $n$ lines. If $P$ is the set of vertices of a regular $n$-gon, no line contains more than $2$ points, so clause 2 must hold; indeed all $\binom{n}{2}$ lines are distinct.

Going deeper

For a student

József Beck proved the theorem in 1983, and it is a standard application of the Szemerédi–Trotter incidence bound. A consequence is a weak form of Dirac's conjecture: some point of $P$ lies on at least $c n$ of the lines determined by $P$, unless most points are collinear. It belongs to the circle of results around the Sylvester–Gallai theorem and the orchard-planting problem, which count lines with few points.