{"schema":"vela.problem-packet.v0.1","problem":176,"statement":"Let $N(k,\\ell)$ be the minimal $N$ such that for any $f:\\{1,\\ldots,N\\}\\to\\{-1,1\\}$ there must exist a $k$-term arithmetic progression $P$ such that\\[ \\left\\lvert \\sum_{n\\in P}f(n)\\right\\rvert\\geq \\ell.\\]Find good upper bounds for $N(k,\\ell)$. Is it true that for any $c>0$ there exists some $C>1$ such that\\[N(k,ck)\\leq C^k?\\]What about\\[N(k,2)\\leq C^k\\]or\\[N(k,\\sqrt{k})\\leq C^k?\\]","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)"}