Hamilton saturated hypergraphs of essentially minimum size (Q1953507)

Summary: For \(1\leq \ell< k\), an \(\ell\)-overlapping cycle is a \(k\)-uniform hypergraph in which, for some cyclic vertex ordering, every edge consists of \(k\) consecutive vertices and every two consecutive edges share exactly \(\ell\) vertices. A \(k\)-uniform hypergraph \(H\) is \(\ell\)-Hamiltonian saturated, \(1\leq \ell\leq k-1\), if \(H\) does not contain an \(\ell\)-overlapping Hamiltonian cycle \(C^{(k)}_n(\ell)\) but every hypergraph obtained from \(H\) by adding one more edge does contain \(C^{(k)}_n(\ell)\). Let \(sat(n,k,\ell)\) be the smallest number of edges in an \(\ell\)-Hamiltonian saturated \(k\)-uniform hypergraph on \(n\) vertices. \textit{L. Clark} and \textit{R. Entringer} [Period. Math. Hung. 14, 57--68 (1983; Zbl 0489.05038)] proved in 1983 that \(sat(n,2,1)=\lceil \tfrac{3n}2\rceil\) and the second author showed recently that \(sat(n,k,k-1)=\Theta(n^{k-1})\) for every \(k\geq2\). In this paper we prove that \(sat(n,k,\ell)=\Theta(n^{\ell})\) for \(\ell=1\) as well as for all \(k\geq5\) and \(\ell\geq0.8k\).