Let $k>r$ and $n$ be sufficiently large in terms of $k$ and $r$. Does there always exist a block $r-(n,k,1)$ design (or Steiner system with parameters $(n,k,r)$), provided the trivial necessary divisibility conditions $\left(\genfrac{}{}{0ex}{}{k-i}{r-i}\right)\mid \left(\genfrac{}{}{0ex}{}{n-i}{r-i}\right)$ are satisfied for every $0\le i<r$?That is, can one find a family of $\left(\genfrac{}{}{0ex}{}{n}{k}\right){\left(\genfrac{}{}{0ex}{}{k}{r}\right)}^{-1}$ many subsets of $\{1,\dots ,n\}$, all of size $k$, such that any $A\subseteq \{1,\dots ,n\}$ of size $r$ is contained in exactly one set in the family?