Forbidding Hamilton cycles in uniform hypergraphs

For $1\le d\le \mathrm{\ell }<k$, we give a new lower bound for the minimum $d$-degree threshold that guarantees a Hamilton $\mathrm{\ell }$-cycle in $k$-uniform hypergraphs. When $k\ge 4$ and $d<\mathrm{\ell }=k-1$, this bound is larger than the conjectured minimum $d$-degree threshold for perfect matchings and thus disproves a well-known conjecture of R"odl and Ruci'nski. Our (simple) construction generalizes a construction of Katona and Kierstead and the space barrier for Hamilton cycles.