Simple Proofs of Classical Theorems in Discrete Geometry via the   Guth--Katz Polynomial Partitioning Technique
  Recently Guth and Katz \cite{GK2} invented, as a step in their nearly complete solution of Erd\H{o}s's distinct distances problem, a new method for partitioning finite point sets in $\R^d$, based on the Stone--Tukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemer\'edi--Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generalizations and extensions, such as the Pach--Sharir bound on the number of incidences with algebraic curves of bounded degree. 
Haim Kaplan and Ji\v{r}\'i Matou\v{s}ek and Micha Sharir
math.CO