Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Coloring the cube with rainbow cycles   For every even positive integer $k\ge 4$ let $f(n,k)$ denote the minimim
number of colors required to color the edges of the $n$-dimensional cube $Q_n$,
so that the edges of every copy of $k$-cycle $C_k$ receive $k$ distinct colors.
Faudree, Gy\'arf\'as, Lesniak and Schelp proved that $f(n,4)=n$ for $n=4$ or
$n>5$. We consider larger $k$ and prove that if $k \equiv 0$ (mod 4), then
there are positive constants $c_1, c_2$ depending only on $k$ such that
$$c_1n^{k/4} < f(n,k) < c_2 n^{k/4}.$$ Our upper bound uses an old construction
of Bose and Chowla of generalized Sidon sets. For $k \equiv 2$ (mod 4), the
situation seems more complicated. For the smallest case k=6 we show that $$n
\le f(n, 6) < n^{1+o(1)}.$$ The upper bound is obtained from Behrend's
construction of a subset of the integers with no three term arithmetic
progression.

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The categories of this paper are: funct-an
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: stat
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: eess
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The categories of this paper are: physics
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: q-fin
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The categories of this paper are: math-ph