Erdős Problem #722

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 \(\binom{k-i}{r-i}\mid \binom{n-i}{r-i}\) are satisfied for every \(0\leq i<r\)? That is, can one find a family of \(\binom{n}{r}\binom{k}{r}^{-1}\) many subsets of \(\{1,\ldots,n\}\), all of size \(k\), such that any \(A\subseteq \{1,\ldots,n\}\) of size \(r\) is contained in exactly one set in the family? This was proved by Keevash [Ke14] for all \((r,k)\).