{"schema":"vela.problem-packet.v0.1","problem":656,"statement":"Let $A\\subset \\mathbb{N}$ be a set with positive upper density. Must there exist an infinite set $B\\subseteq A$ and integer $t$ such that\\[\\{b_1+b_2: b_1\\neq b_2\\in B\\}+t\\subseteq A?\\]","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)"}