{"schema":"vela.problem-packet.v0.1","problem":777,"statement":"If $\\mathcal{F}$ is a family of subsets of $\\{1,\\ldots,n\\}$ then we write $G_{\\mathcal{F}}$ for the graph on $\\mathcal{F}$ where $A\\sim B$ if $A$ and $B$ are comparable - that is, $A\\subseteq B$ or vice versa.Is it true that, if $\\epsilon&#62;0$ and $n$ is sufficiently large, whenever $m\\leq (2-\\epsilon)2^{n/2}$ the graph $G_\\mathcal{F}$ has $<2^{n}$ many edges?Is it true that if $G_{\\mathcal{F}}$ has $\\geq cm^2$ edges then $m\\ll_c 2^{n/2}$?Is it true that, for any $\\epsilon>0$, there exists some $\\delta&#62;0$ such that if there are $&#62;m^{2-\\delta}$ edges then $m&#60;(2+\\epsilon)^{n/2}$?","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)"}