Hitting times for Shamir’s problem

For fixed r≥ 3 and n divisible by r, let H= Hʳn,M be the random M-edge r-graph on V=\1,… ,n\; that is, H is chosen uniformly from the M-subsets of K:=V r (:= $-subsets of $V\). Shamir's Problem (circa 1980) asks, roughly, for what $M=M(n)$ is 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>Cᵣnlog n. More recently the author proved the asymptotically correct version of that result: for fixed $C> 1/r$ and M> Cnlog n, P( H ~contains a perfect matching)→ 1 \,\,\, asn→∞. The present work completes a proof, begun in that recent paper, of the definitive "hitting time" statement: Theorem. If A₁, … ~ is a uniform permutation of K, Hₜ=₁ Aₜ\, and \[ T=min\{t:A_1∪ ⋯∪ A_t=V\}, \] then P( HT ~contains a perfect matching)→ 1 \,\,\, asn→∞.