{"paper":{"title":"Anti-van der Waerden numbers of 3-term arithmetic progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alex Schulte, Michael Young, Zhanar Berikkyzy","submitted_at":"2016-04-29T13:12:32Z","abstract_excerpt":"The \\emph{anti-van der Waerden number}, denoted by $aw([n],k)$, is the smallest $r$ such that every exact $r$-coloring of $[n]$ contains a rainbow $k$-term arithmetic progression. Butler et. al. showed that $\\lceil \\log_3 n \\rceil + 2 \\le aw([n],3) \\le \\lceil \\log_2 n \\rceil + 1$, and conjectured that there exists a constant $C$ such that $aw([n],3) \\le \\lceil \\log_3 n \\rceil + C$. In this paper, we show this conjecture is true by determining $aw([n],3)$ for all $n$. We prove that for $7\\cdot 3^{m-2}+1 \\leq n \\leq 21 \\cdot 3^{m-2}$, \\[ aw([n],3)=\\left\\{\\begin{array}{ll} m+2, & \\mbox{if $n=3^m$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.08819","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}