Extremal numbers of hypergraph suspensions of even cycles

For fixed k≥2, determining the order of magnitude of the number of edges in an n-vertex bipartite graph not containing C2k, 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 maximum number of triples in an n-vertex triple system which does not contain a C6 in the link of any vertex, has order of magnitude n7/3. Additionally, we construct new families of dense C6-free bipartite graphs with n vertices and n4/3 edges in order of magnitude.