Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Long Alternating Paths Exist   Let $P$ be a set of $2n$ points in convex position, such that $n$ points are
colored red and $n$ points are colored blue. A non-crossing alternating path on
$P$ of length $\ell$ is a sequence $p_1, \dots, p_\ell$ of $\ell$ points from
$P$ so that (i) all points are pairwise distinct; (ii) any two consecutive
points $p_i$, $p_{i+1}$ have different colors; and (iii) any two segments $p_i
p_{i+1}$ and $p_j p_{j+1}$ have disjoint relative interiors, for $i \neq j$.
  We show that there is an absolute constant $\varepsilon > 0$, independent of
$n$ and of the coloring, such that $P$ always admits a non-crossing alternating
path of length at least $(1 + \varepsilon)n$. The result is obtained through a
slightly stronger statement: there always exists a non-crossing bichromatic
separated matching on at least $(1 + \varepsilon)n$ points of $P$. This is a
properly colored matching whose segments are pairwise disjoint and intersected
by common line. For both versions, this is the first improvement of the easily
obtained lower bound of $n$ by an additive term linear in $n$. The best known
published upper bounds are asymptotically of order $4n/3+o(n)$.

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The categories of this paper are: quant-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: q-alg
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The categories of this paper are: nlin
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The categories of this paper are: nucl-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: math-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: q-fin
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-th
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The categories of this paper are: solv-int