  We show that for all $k\geq 4$, $\varepsilon >0$, and $n$ sufficiently large,
every $k$-uniform hypergraph on $n$ vertices in which each set of $k-3$
vertices is contained in at least $(5/8 + \varepsilon) \binom{n}{3}$ edges
contains a tight Hamilton cycle. This is asymptotically best possible.
