{"schema":"vela.problem-packet.v0.1","problem":179,"statement":"Let $1\\leq k<\\ell$ be integers and define $F_k(N,\\ell)$ to be minimal such that every set $A\\subset \\mathbb{N}$ of size $N$ which contains at least $F_k(N,\\ell)$ many $k$-term arithmetic progressions must contain an $\\ell$-term arithmetic progression. Find good upper bounds for $F_k(N,\\ell)$. Is it true that\\[F_3(N,4)=o(N^2)?\\]Is it true that for every $\\ell>3$\\[\\lim_{N\\to \\infty}\\frac{\\log F_3(N,\\ell)}{\\log N}=2?\\]","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)"}