{"schema":"vela.problem-packet.v0.1","problem":721,"statement":"Let $W(3,k)$ be the van der Waerden number defined as the minimum $n$ such that in any red/blue colouring of $\\{1,\\ldots,n\\}$ there exists either a red $3$-term arithmetic progression or a blue $k$-term arithmetic progression.Give reasonable bounds for $W(3,k)$. In particular, give any non-trivial lower bounds for $W(3,k)$ and prove that $W(3,k) < \\exp(k^c)$ for some constant $c<1$.","status":"solved","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A171081","name":"Van der Waerden numbers w(3, n).","terms":"9,18,22,32,46,58,77,97,114,135,160,186,218,238,279,312,349","url":"https://oeis.org/A171081"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}