Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Tur\'an number of Berge book hypergraphs   Given a graph $G$, a Berge copy of $G$ is a hypergraph obtained by enlarging
the edges arbitrarily. Gy\H ori in 2006 showed that for $r=3$ or $r=4$, an
$r$-uniform $n$-vertex Berge triangle-free hypergraph has at most $\lfloor
n^2/8(r-2)\rfloor$ hyperedges if $n$ is large enough, and this bound is sharp.
  The book graph $B_t$ consists of $t$ triangles sharing an edge. Very
recently, Ghosh, Gy\H{o}ri, Nagy-Gy\"orgy, Paulos, Xiao and Zamora showed that
a 3-uniform $n$-vertex Berge $B_t$-free hypergraph has at most $n^2/8+o(n^2)$
hyperedges if $n$ is large enough. They conjectured that this bound can be
improved to $\lfloor n^2/8\rfloor$.
  We prove this conjecture for $t=2$ and disprove it for $t>2$ by proving the
sharp bound $\lfloor n^2/8\rfloor+(t-1)^2$. We also consider larger uniformity
and determine the largest number of Berge $B_t$-free $r$-uniform hypergraphs
besides an additive term $o(n^2)$. We obtain a similar bound if the Berge
$t$-fan ($t$ triangles sharing a vertex) is forbidden.

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The categories of this paper are: cond-mat
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The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
1
The categories of this paper are: cs
18
The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
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The categories of this paper are: solv-int
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The categories of this paper are: q-bio
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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-th
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The categories of this paper are: bayes-an
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The categories of this paper are: stat
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The categories of this paper are: nucl-th