Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the number of touching pairs in a set of planar curves   Given a set of planar curves (Jordan arcs), each pair of which meets --
either crosses or touches -- exactly once, we establish an upper bound on the
number of touchings. We show that such a curve family has $O(t^2n)$ touchings,
where $t$ is the number of faces in the curve arrangement that contains at
least one endpoint of one of the curves. Our method relies on finding special
subsets of curves called quasi-grids in curve families; this gives some
structural insight into curve families with a high number of touchings.

18
The categories of this paper are: econ
27
The categories of this paper are: comp-gas
11
The categories of this paper are: hep-lat
21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
28
The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph
17
The categories of this paper are: q-fin
24
The categories of this paper are: dg-ga
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
35
The categories of this paper are: mtrl-th