{"schema":"vela.problem-packet.v0.1","problem":817,"statement":"Let $k\\geq 3$ and define $g_k(n)$ to be the minimal $N$ such that $\\{1,\\ldots,N\\}$ contains some $A$ of size $\\lvert A\\rvert=n$ such that\\[\\langle A\\rangle = \\left\\{\\sum_{a\\in A}\\epsilon_aa: \\epsilon_a\\in \\{0,1\\}\\right\\}\\]contains no non-trivial $k$-term arithmetic progression. Estimate $g_k(n)$. In particular, is it true that\\[g_3(n) \\gg 3^n?\\]","status":"open","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)"}