  For fixed $r\geq 3$ and $n$ divisible by $r$, let ${\mathcal H}={\mathcal
H}^r_{n,M}$ be the random $M$-edge $r$-graph on $V=\{1,\ldots ,n\}$; that is,
${\mathcal H}$ is chosen uniformly from the $M$-subsets of ${\mathcal K}:={V
\choose r}$ ($:= \{\mbox{$r$-subsets of $V$}\}$). Shamir's Problem (circa 1980)
asks, roughly, for what $M=M(n)$ is ${\mathcal H}$ likely to contain a perfect
matching (that is, $n/r$ disjoint $r$-sets)?
  In 2008 Johansson, Vu and the author showed that this is true for $M>C_rn\log
n$. More recently the author proved the asymptotically correct version of that
result: for fixed $C> 1/r$ and $M> Cn\log n$, $P({\mathcal H} ~\mbox{contains a
perfect matching})\rightarrow 1 \,\,\, \mbox{as $n\rightarrow\infty$}.$
  The present work completes a proof, begun in that recent paper, of the
definitive "hitting time" statement:
  $\mbox{Theorem.}$ If $A_1, \ldots ~$ is a uniform permutation of ${\mathcal
K}$, ${\mathcal H}_t=\{A_1\dots A_t\}$, and \[ T=\min\{t:A_1\cup \cdots\cup
A_t=V\}, \] then $P({\mathcal H}_T ~\mbox{contains a perfect
matching})\rightarrow 1 \,\,\, \mbox{as $n\rightarrow\infty$}$.
