TITLE: Szemerédi–Trotter theorem
QUESTION [7 upvotes]: Szemerédi–Trotter theorem asserts that given $n$ points and $m$ lines in the plane, the number of incidences (i.e., the number of point-line pairs, such that the point lies on the line) is:
$O((mn)^{\frac{2}{3}}+m+n).$
I was wondering how many different proofs are known for this theorem? Is there any survey or references for these proofs? I know there are two proofs here https://www.cs.princeton.edu/~zdvir/papers/Dvir-survey.pdf

REPLY [2 votes]: I can think of at least three proofs:

The original cell decomposition proof of Szemerédi-Trotter
The proof via the Crossing Lemma given by Székely
The proof using the polynomial ham sandwich theorem

Adam Sheffer has an early draft of a book on incidence theory and the polynomial method, which mentions all three proofs.