Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Tur\'{a}n numbers of the complete $4$-graphs   The Tur\'{a}n number $T(n,\alpha+1,r)$ is the minimum number of edges in an
$n$-vertex $r$-graph whose independence number does not exceed $\alpha$. For
each $r\geq 2$, there exists $t_*(r)$ such that $T(n,\alpha+1,r) = t_*(r) \:
n^r \: \alpha^{1-r} \: (1+o(1))$ as $\alpha / r \to\infty$ and $n / \alpha
\to\infty$. It is known that $t_*(2) = 1/2$, and the conjectured value of
$t_*(3)$ is $2/3$. We prove that $t_*(4) < 0.706335\:$.

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The categories of this paper are: hep-th
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The categories of this paper are: hep-ph
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The categories of this paper are: physics
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The categories of this paper are: comp-gas
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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: eess
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph
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The categories of this paper are: adap-org
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: solv-int
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The categories of this paper are: quant-ph