Dirac-type results for loose Hamilton cycles in uniform hypergraphs

A classic result of G.A. Dirac in graph theory asserts that for n⩾3 every n-vertex graph with minimum degree at least n/2 contains a spanning (so-called Hamilton) cycle. G.Y. Katona and H.A. Kierstead suggested a possible extension of this result for k-uniform hypergraphs. There a Hamilton cycle of an n-vertex hypergraph corresponds to an ordering of the vertices such that every k consecutive (modulo n) vertices in the ordering form an edge. Moreover, the minimum degree is the minimum (k−1)-degree, i.e. the minimum number of edges containing a fixed set of k−1 vertices. V. Rödl, A. Ruciński, and E. Szemerédi verified (approximately) the conjecture of Katona and Kierstead and showed that every n-vertex, k-uniform hypergraph with minimum (k−1)-degree (1/2+o(1))n contains such a tight Hamilton cycle. We study the similar question for Hamilton ℓ-cycles. A Hamilton ℓ-cycle in an n-vertex, k-uniform hypergraph (1⩽ℓ