  For fixed $k\ge 2$, determining the order of magnitude of the number of edges
in an $n$-vertex bipartite graph not containing $C_{2k}$, the cycle of length
$2k$, is a long-standing open problem. We consider an extension of this problem
to triple systems. In particular, we prove that the number of triples in an
$n$-vertex triple system which does not contain a $C_6$ in the link of any
vertex, has order of magnitude $n^{7/3}$. Additionally, we construct new
families of dense $C_6$-free bipartite graphs with $n$ vertices and $n^{4/3}$
edges in order of magnitude.
