Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Edge-Colored Saturation Problems   Let $\mathcal{C}$ be a family of edge-colored graphs. A $t$-edge colored
graph $G$ is $(\mathcal{C}, t)$-saturated if $G$ does not contain any graph in
$\mathcal{C}$ but the addition of any edge in any color in $[t]$ creates a copy
of some graph in $\mathcal{C}$. Similarly to classical saturation functions,
define $\mathrm{sat}_t(n, \mathcal{C})$ to be the minimum number of edges in a
$(\mathcal{C},t)$ saturated graph. Let $\mathcal{C}_r(H)$ be the family
consisting of every edge-colored copy of $H$ which uses exactly $r$ colors.
  In this paper we consider a variety of colored saturation problems. We
determine the order of magnitude for $\mathrm{sat}_t(n, \mathcal{C}_r(K_k))$
for all $r$, showing a sharp change in behavior when $r\geq \binom{k-1}{2}+2$.
A particular case of this theorem proves a conjecture of Barrus, Ferrara,
Vandenbussche, and Wenger. We determine $\mathrm{sat}_t(n, \mathcal{C}_2(K_3))$
exactly and determine the extremal graphs. Additionally, we document some
interesting irregularities in the colored saturation function.

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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: chem-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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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: cs
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The categories of this paper are: plasm-ph
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The categories of this paper are: physics
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The categories of this paper are: nlin
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The categories of this paper are: quant-ph