{"schema":"vela.problem-packet.v0.1","problem":139,"statement":"Let $r_k(N)$ be the size of the largest subset of $\\{1,\\ldots,N\\}$ which does not contain a non-trivial $k$-term arithmetic progression. Prove that $r_k(N)=o(N)$.","status":"solved","seam":"raw","closureRoutes":[],"obligations":[],"attestations":[],"attempts":[],"velaLean":[],"oeis":[{"id":"A003002","name":"Size of the largest subset of the numbers [1...n] which does not contain a 3-term arithmetic progression.","terms":"0,1,2,2,3,4,4,4,4,5,5,6,6,7,8,8,8,8,8,8,9,9,9,9,10,10,11,11,11,11,12,12,13,13,13,13,14,14,14,14,15,16,16,16,16,16,16,16,","url":"https://oeis.org/A003002"},{"id":"A003003","name":"Size of the largest subset of the numbers [1...n] which doesn't contain a 4-term arithmetic progression.","terms":"1,2,3,3,4,5,5,6,7,8,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15,16,17,17,18,18,18,19,20,20,20,21,21,21,22,22,22,23,23,24","url":"https://oeis.org/A003003"},{"id":"A003004","name":"Size of the largest subset of the numbers [1..n] which does not contain a 5-term arithmetic progression.","terms":"1,2,3,4,4,5,6,7,8,8,9,10,11,12,12,13,14,15,16,16,16,16,16,17,18,18,19,20,21,21,22,22,23,24,24,25,26,27,28,28,29,30,31,32","url":"https://oeis.org/A003004"},{"id":"A003005","name":"Size of the largest subset of the numbers [1..n] which doesn't contain a 6-term arithmetic progression.","terms":"1,2,3,4,5,5,6,7,8,9,9,10,11,12,13,13,14,15,16,17,17,18,19,20,21,22,22,22,23,23,23,24,25,25,26,27,28,28,29,30,31,31,31,32","url":"https://oeis.org/A003005"}],"generated_note":"derived view; the signed event log is the source of truth (vela check / vela reproduce)"}