  In this paper we study the maximum number of hyperedges which may be in an
$r$-uniform hypergraph under the restriction that no pair of vertices has more
than $t$ Berge paths of length $k$ between them. When $r=t=2$, this is the
even-cycle problem asking for $\mathrm{ex}(n, C_{2k})$. We extend results of
F\"uredi and Simonovits and of Conlon, who studied the problem when $r=2$. In
particular, we show that for fixed $k$ and $r$, there is a constant $t$ such
that the maximum number of edges can be determined in order of magnitude.
