Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Beyond the Richter-Thomassen Conjecture   If two closed Jordan curves in the plane have precisely one point in common,
then it is called a {\em touching point}. All other intersection points are
called {\em crossing points}. The main result of this paper is a Crossing Lemma
for closed curves: In any family of $n$ pairwise intersecting simple closed
curves in the plane, no three of which pass through the same point, the number
of crossing points exceeds the number of touching points by a factor of at
least $\Omega((\log\log n)^{1/8})$.
  As a corollary, we prove the following long-standing conjecture of Richter
and Thomassen: The total number of intersection points between any $n$ pairwise
intersecting simple closed curves in the plane, no three of which pass through
the same point, is at least $(1-o(1))n^2$.

29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
27
The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
11
The categories of this paper are: hep-lat
14
The categories of this paper are: math-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: econ