{"schema":"vela.problem-packet.v0.1","problem":1183,"statement":"Let $f(n)$ be maximal such that in any $2$-colouring of the subsets of $\\{1,\\ldots,n\\}$ there is always a monochromatic family of at least $f(n)$ sets which is closed under taking unions and intersections. Estimate $f(n)$.Let $F(n)$ be defined similarly, except that we only require the family be closed under taking unions. Estimate $F(n)$. In particular, is it true that $F(n)\\geq n^{\\omega(n)}$ for some $\\omega(n)\\to \\infty$ as $n\\to \\infty$, and $F(n)<(1+o(1))^n$?","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)"}