Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Rainbow polygons for colored point sets in the plane   Given a colored point set in the plane, a perfect rainbow polygon is a simple
polygon that contains exactly one point of each color, either in its interior
or on its boundary. Let $\operatorname{rb-index}(S)$ denote the smallest size
of a perfect rainbow polygon for a colored point set $S$, and let
$\operatorname{rb-index}(k)$ be the maximum of $\operatorname{rb-index}(S)$
over all $k$-colored point sets in general position; that is, every $k$-colored
point set $S$ has a perfect rainbow polygon with at most
$\operatorname{rb-index}(k)$ vertices. In this paper, we determine the values
of $\operatorname{rb-index}(k)$ up to $k=7$, which is the first case where
$\operatorname{rb-index}(k)\neq k$, and we prove that for $k\ge 5$, \[
\frac{40\lfloor (k-1)/2 \rfloor -8}{19} %Birgit:
\leq\operatorname{rb-index}(k)\leq 10 \bigg\lfloor\frac{k}{7}\bigg\rfloor + 11.
\] Furthermore, for a $k$-colored set of $n$ points in the plane in general
position, a perfect rainbow polygon with at most $10 \lfloor\frac{k}{7}\rfloor
+ 11$ vertices can be computed in $O(n\log n)$ time.

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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: nucl-ex
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
5
The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
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The categories of this paper are: stat
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The categories of this paper are: quant-ph
12
The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
0
The categories of this paper are: hep-ph