Hitting times for Shamir’s problem

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