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{\bf Felix Lazebnik and Jacques Verstra\" ete}
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{\bf On Hypergraphs of Girth Five}
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In this paper, we study $r$-uniform hypergraphs ${\cal H}$ without
cycles of length less than five, employing the definition of a
hypergraph cycle due to Berge. In particular, for $r = 3$, we show
that if ${\cal H}$ has $n$ vertices and a maximum number of edges,
then
$$|{\cal H}|={\textstyle 1\over6}n^{3/2} + o(n^{3/2}).$$

This also asymptotically determines  the generalized Tur\' an
number $T_{3}(n,8,4)$. Some results are based on our bounds for
the maximum size of Sidon-type sets in $\Bbb{Z}_{n}$.

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