Random threshold conjecture for Steiner designs

Let $t,k,\lambda\in\mathbb N$ with $k>t$, let $V$ be an $n$-element set satisfying the divisibility conditions $\binom{k-i}{t-i}\mid\lambda\binom{n-i}{t-i}$ for every $i\in\{0\}\cup[t-1]$, and let $\mathcal A$ consist of independently selected $k$-sets from $\binom Vk$, each included with probability $p$. A $t$-$(n,k,\lambda)$ design is a pair $(V,\mathcal B)$ with $\mathcal B\subseteq\binom Vk$ such that every $t$-set is contained in exactly $\lambda$ members of $\mathcal B$. Random design threshold conjecture. If $p=\omega(\log n/n^{k-t})$, then with probability tending to one there exists a $t$-$(n,k,\lambda)$ design $(V,\mathcal B)$ with $\mathcal B\subseteq\mathcal A$. The conjecture asserts that the threshold is governed by the requirement that every $t$-set be covered at least $\lambda$ times.