Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
13
The categories of this paper are: quant-ph
18
The categories of this paper are: econ
1
The categories of this paper are: cs
9
The categories of this paper are: nlin
16
The categories of this paper are: stat
35
The categories of this paper are: mtrl-th
23
The categories of this paper are: solv-int
17
The categories of this paper are: q-fin
12
The categories of this paper are: nucl-th
4
The categories of this paper are: cond-mat
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
21
The categories of this paper are: adap-org