Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Profile of Multiplicities of Complete Subgraphs   Let $G$ be a $2$-coloring of a complete graph on $n$ vertices, for
sufficiently large $n$. We prove that $G$ contains at least $n^{(\frac{1}{4} -
o(1))\log n}$ monochromatic complete subgraphs of size $r$, where \[ 0.3\log n
< r < 0.7\log n. \] The previously known lower bound on the total number of
monochromatic complete subgraphs, due to Sz\'{e}kely was $n^{0.1576\log n}$. We
also prove that $G$ contains at least $n^{\frac{1}{7} \log n} $ monochromatic
complete subgraphs of size $\frac{1}{2}\log n$.
  If furthermore one assumes that the largest monochromatic complete subgraph
in $G$ is of size $(\frac{1}{2} + o(1))\log n$ (it is a well known open
question whether such graphs exist), then for every constant $0 \le c \le
\frac{1}{2}$ we determine (up to low order terms) the number of monochromatic
complete subgraphs of size $c \log n$. We do so by proving a lower bound that
matches (up to low order terms) a previous upper bound of Sz\'{e}kely. For
example, the number of monochromatic complete subgraphs of size $\frac{1}{2}
\log n$ is $n^{\frac{1}{8}(4 - \log e \pm o(1))\log n} \simeq n^{0.32 \log n}$.

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The categories of this paper are: nucl-th
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The categories of this paper are: quant-ph
31
The categories of this paper are: ao-sci
5
The categories of this paper are: gr-qc
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The categories of this paper are: patt-sol
21
The categories of this paper are: adap-org
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The categories of this paper are: supr-con
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The categories of this paper are: q-alg
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: stat
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The categories of this paper are: solv-int
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The categories of this paper are: econ
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The categories of this paper are: hep-ex