Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Coloring points with respect to squares   We consider the problem of $2$-coloring geometric hypergraphs. Specifically,
we show that there is a constant $m$ such that any finite set of points in the
plane $\mathcal{S} \subset {\mathbb R}^2$ can be $2$-colored such that every
axis-parallel square that contains at least $m$ points from $\mathcal{S}$
contains points of both colors. Our proof is constructive, that is, it provides
a polynomial-time algorithm for obtaining such a $2$-coloring. By affine
transformations this result immediately applies also when considering
$2$-coloring points with respect to homothets of a fixed parallelogram.

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The categories of this paper are: patt-sol
9
The categories of this paper are: nlin
34
The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: ao-sci
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The categories of this paper are: eess
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The categories of this paper are: econ
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: q-bio
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The categories of this paper are: q-fin
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The categories of this paper are: gr-qc
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The categories of this paper are: acc-phys
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The categories of this paper are: stat
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The categories of this paper are: nucl-ex
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The categories of this paper are: atom-ph