Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Tur\'an number of Berge-K_4 in triple systems   A Berge-$K_4$ in a triple system is a configuration with four vertices
$v_1,v_2,v_3,v_4$ and six distinct triples $\{e_{ij}: 1\le i< j \le 4\}$ such
that $\{v_i,v_j\}\subset e_{ij}$ for every $1\le i<j\le 4$. We denote by
$\cal{B}$ the set of Berge-$K_4$ configurations. A triple system is
$\cal{B}$-free if it does not contain any member of $\cal{B}$. We prove that
the maximum number of triples in a $\cal{B}$-free triple system on $n\ge 6$
points is obtained by the balanced complete $3$-partite triple system: all
triples $\{abc: a\in A, b\in B, c\in C\}$ where $A,B,C$ is a partition of $n$
points with $$\left\lfloor{n\over 3}\right\rfloor=|A|\le |B|\le
|C|=\left\lceil{n\over 3}\right\rceil.$$

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The categories of this paper are: comp-gas
18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: eess
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The categories of this paper are: solv-int
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The categories of this paper are: adap-org
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
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The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
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The categories of this paper are: chem-ph
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The categories of this paper are: stat