Growth order for the size of smallest hamiltonian chain saturated uniform hypergraphs

We say that a hypergraph H is hamiltonian chain saturated if H does not contain a hamiltonian chain but by adding any new edge we create a hamiltonian chain in H. In this paper, for each k≥3, we establish the right order of magnitude nk−1 for the size of the smallest k-uniform hamiltonian chain saturated hypergraph. This solves an open problem of G.Y. Katona.