{"schema":"vela.problem-packet.v0.1","problem":12,"statement":"Let $A$ be an infinite set such that there are no distinct $a,b,c\\in A$ such that $a\\mid (b+c)$ and $b,c>a$. Is there such an $A$ with\\[\\liminf \\frac{\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert}{N^{1/2}}>0?\\]Does there exist some absolute constant $c>0$ such that there are always infinitely many $N$ with\\[\\lvert A\\cap\\{1,\\ldots,N\\}\\rvert<N^{1-c}?\\]Is it true that\\[\\sum_{n\\in A}\\frac{1}{n}<\\infty?\\]","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[{"verdict":"faithful","attestedBy":"reviewer:will-blair","formalRef":"erdos_12.parts.i.lean","targetFinding":"vf_caaa525f7259eff1"}],"attempts":[],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}