{"schema":"vela.problem-packet.v0.1","problem":732,"statement":"Call a sequence $1< X_1\\leq \\cdots \\leq X_m\\leq n$ block-compatible if there is a pairwise balanced block design $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ such that $\\lvert A_i\\rvert=X_i$ for $1\\leq i\\leq m$. (A pairwise block design means that every pair in $\\{1,\\ldots,n\\}$ is contained in exactly one of the $A_i$.)Are there necessary and sufficient conditions for $(X_i)$ to be block-compatible?Is there some constant $c>0$ such that for all large $n$ there are\\[\\geq \\exp(c n^{1/2}\\log n)\\]many block-compatible sequences for $\\{1,\\ldots,n\\}$?","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)"}