  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 $M>C_rn\log
n$. The present paper has two purposes. First, it establishes the
asymptotically correct version of the 2008 result:
  Theorem 1. For fixed $\epsilon>0$ and $M> (1+\epsilon)(n/r)\log n$,
$P({\mathcal H} ~\mbox{contains a perfect matching})\rightarrow 1 $ as
$n\rightarrow\infty$.
  Second, it begins a proof of the definitive ``hitting time" statement:
  Theorem 2. If $A_1, \ldots ~$ is a uniform permutation of ${\mathcal K}$,
${\mathcal H}_t=\{A_1,\ldots ,A_t\}$, and $T=\min\{t:A_1\cup \cdots\cup
A_t=V\},$ then $P({\mathcal H}_T ~\mbox{contains a perfect
matching})\rightarrow 1 $ as $n\rightarrow\infty$.
  It is shown here that Theorem 2 follows from a conditional version of Theorem
1 that will be proved elsewhere. The key ideas in that proof are similar to
those for Theorem 1, but the argument is a longer story, and it has seemed best
to give the present separate proof of Theorem 1, in which those ideas may
appear more clearly.
