{"schema":"vela.problem-packet.v0.1","problem":722,"statement":"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 $\\binom{n}{k}\\binom{k}{r}^{-1}$ 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?","status":"solved","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}