{"schema":"vela.problem-packet.v0.1","problem":535,"statement":"Let $r\\geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\\{1,\\ldots,N\\}$ such that no subset of size $r$ has the same pairwise greatest common divisor between all elements. Estimate $f_r(N)$.","status":"open","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[{"id":"att_5b90d6b0e9df810f","kind":"partial_proof","claim":"Erdos 535 remains open; I give a rigorous conditional reduction (strong Omega-sunflower bound g_r(k)<=c_r^k => f_r(N)<=N^{2 ln c_r/loglog N}), with the layer/Poisson counting mechanism proven analytic","grade":"partial_proof","gateStatus":"needs_verification","superseded":false}],"velaLean":[],"oeis":[],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}