Robust thresholds: Counting triangle factors and A shorter proof of the robust Corrádi-Hajnal Theorem

We show that the distribution of perfect matchings in an $r$-uniform hypergraph on $n$ vertices induced by the edges of an $r$-partite super-regular graph is $O(1/n^{r-1})$-spread. This yields a short proof of the robust Corr\'adi-Hajnal Theorem recently obtained by Allen et al., that there is $C>0$ such that, for a graph $G$ with minimum degree $2n/3$, a random subgraph of $G$ where each edge is retained with probability $C(\log n)^{1/3}n^{-2/3}$ contains a triangle factor with high probability. We also show that the number of triangle factors in a graph with minimum degree $2n/3$ is at least $(cn)^{2n/3}$ for a constant $c>0$, addressing a question of Allen et al.