{"schema":"vela.problem-packet.v0.1","problem":808,"statement":"Let $c,\\epsilon>0$ and $n$ be sufficiently large. If $A\\subset \\mathbb{N}$ has $\\lvert A\\rvert=n$ and $G$ is any graph on $A$ with at least $n^{1+c}$ edges then\\[\\max(\\lvert A+_GA\\rvert,\\lvert A\\cdot_G A\\rvert) \\geq \\lvert A\\rvert^{1+c-\\epsilon},\\]where\\[A+_GA = \\{ a+b : (a,b)\\in G\\}\\]and similarly for $A\\cdot_GA$.","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)"}