Constructing sparsest $\ell$-hamiltonian saturated $k$-uniform hypergraphs for a wide range of $\ell$

Abstract

Given $k\ge 3$ and $1\le \ell <k$, an $(\ell ,k)$-cycle is one in which consecutive edges, each of size $k$, overlap in exactly $\ell$ vertices. We study the smallest number of edges in $k$-uniform $n$-vertex hypergraphs which do not contain hamiltonian $(\ell ,k)$-cycles, but once a new edge is added, such a cycle is promptly created. It has been conjectured that this number is of order $n^{\ell }$ and confirmed for $\ell \in \{1,k/2,k-1\}$, as well as for the upper range $0.8k\le \ell \le k-1$. Here we extend the validity of this conjecture to the lower-middle range $(k-1)/3\le \ell <(k-1)/2$.

This is a revised version (we fill a gap in the proof of Lemma 10)