The threshold conjecture for Steiner systems in random hypergraphs

Let $K_k^n$ be the complete $k$-uniform hypergraph on $n$ vertices, let $\mathcal{H}_k(n;p)$ be the random $k$-uniform hypergraph obtained by retaining each edge independently with probability $p$, and let an $(n,k,k-1)$-Steiner system be a collection of $k$-edges in which every $(k-1)$-subset is contained in exactly one edge. Steiner-system threshold conjecture. The threshold for the appearance of an $(n,k,k-1)$-Steiner system in $\mathcal{H}_k(n;p)$ is

$$\Theta\left(\frac{\log n}{n}\right).$$

The lower bound comes from the requirement that no facet, meaning a $(k-1)$-subset, remains uncovered. The conjecture is motivated by the analogous threshold for perfect matchings in hypergraphs, where the obstruction from isolated vertices determines the threshold; the paper notes that its methods only yield a much weaker upper bound.