The threshold conjecture for fractional decompositions of 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 a fractional $K_k$-decomposition of a $k$-uniform hypergraph $H$ be a weight function $w:H\to[0,1]$ such that, for every facet $f\in K_{k-1}(H)$,

$$\sum_{f\subseteq e\in H}w(e)=1.$$

Fractional-decomposition threshold conjecture. The threshold for the appearance of fractional $K_k$-decompositions in $\mathcal{H}_k(n;p)$ is

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

The threshold is bounded below by the disappearance of uncovered facets and above by the threshold for Steiner systems. The conjecture is motivated by the known hitting-time result for fractional matchings, while the corresponding threshold and hitting-time statements for these fractional decompositions remain open.