Szemerédi-Trotter theorem

The number of incidences of a set of $n$ points and a set of $m$ lines in the real plane ${ℝ}^2$ is

$$I=O(n+m+{(nm)}^{\frac{2}{3}}).$$

Proof. Let’s consider the points as vertices of a graph, and connect two vertices by an edge if they are adjacent on some line. Then the number of edges is $e=I-m$. If then we are done. If $e\ge 4n$ then by crossing lemma

$$m^2\ge \mathrm{cr}(G)\ge \frac{1}{64}\frac{{(I-m)}^3}{n^2},$$

and the theorem follows.

Recently, Tóth[1] extended the theorem to the complex plane ${ℂ}^2$. The proof is difficult.