Hamilton Saturated Hypergraphs of Essentially Minimum Size

For 1 ≤ ` < k, an `-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 ` vertices. A k-uniform hypergraph H is `-Hamiltonian saturated, 1 ≤ ` ≤ k − 1, if H does not contain an `-overlapping Hamiltonian cycle C (k) n (`) but every hypergraph obtained from H by adding one more edge does contain C (k) n (`). Let sat(n, k, `) be the smallest number of edges in an `-Hamiltonian saturated k-uniform hypergraph on n vertices. Clark and Entringer proved in 1983 that sat(n, 2, 1) = d 3n 2 e. In this talk we prove that sat(n, k, `) = Θ(n`) for ` = 1 as well as for all k ≥ 5 and ` ≥ 0.8k.