Zoltan Furedi: On 2-connected hypergraphs with no long cycles

We give an upper bound for the maximum number of edges in an n-vertex
2-connected r-uniform hypergraph with no Berge cycle of length k or
greater, where
   $n\geq k \geq 4r\geq 12$.
For n large with respect to r and k, this bound is  sharp and is
significantly stronger than the bound without restrictions on
connectivity. It turned out that it is simpler to prove the bound for
the broader class of Sperner families where the size of each set is at
most r. For such families, our bound is sharp for all   $n\geq k\geq
r\geq 3$.