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 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 maximum 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. (c) 2024 Elsevier Ltd. All rights reserved.