{"schema":"vela.problem-packet.v0.1","problem":665,"statement":"A pairwise balanced design for $\\{1,\\ldots,n\\}$ is a collection of sets $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ such that $2\\leq \\lvert A_i\\rvert <n$ and every pair of distinct elements $x,y\\in \\{1,\\ldots,n\\}$ is contained in exactly one $A_i$.Is there a constant $C>0$ and, for all large $n$, a pairwise balanced design such that\\[\\lvert A_i\\rvert &#62; n^{1/2}-C\\]for all $1\\leq i\\leq m$?","status":"open","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)"}