  Let $Y_{3,2}$ be the $3$-uniform hypergraph with two edges intersecting in
two vertices. Our main result is that any $n$-vertex 3-uniform hypergraph with
at least $\binom{n}{3} - \binom{n-m+1}{3} + o(n^3)$ edges contains a collection
of $m$ vertex-disjoint copies of $Y_{3,2}$, for $m\le n/7$. The bound on the
number of edges is asymptotically best possible. This problem generalizes the
Matching Conjecture of Erd\H{o}s. We then use this result combined with the
absorbing method to determine the asymptotically best possible minimum
$(k-3)$-degree threshold for $\ell$-Hamiltonicity in $k$-graphs, where $k\ge 7$
is odd and $\ell=(k-1)/2$. Moreover, we give related results on $ Y_{k,b}
$-tilings and Hamilton $ \ell $-cycles with $ d $-degree for some other values
of $ k,\ell,d $.
