{"paper":{"title":"A New Lower Bound for van der Waerden Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jay Cummings, Thomas Blankenship, Vladislav Taranchuk","submitted_at":"2017-05-26T18:08:57Z","abstract_excerpt":"In this paper we prove a new recurrence relation on the van der Waerden numbers, $w(r,k)$. In particular, if $p$ is a prime and $p\\leq k$ then $w(r, k) > p \\cdot \\left(w\\left(r - \\left\\lceil \\frac{r}{p}\\right\\rceil, k\\right) -1\\right)$. This recurrence gives the lower bound $w(r, p+1) > p^{r-1}2^p$ when $r \\leq p$, which generalizes Berlekamp's theorem on 2-colorings, and gives the best known bound for a large interval of $r$. The recurrence can also be used to construct explicit valid colorings, and it improves known lower bounds on small van der Waerden numbers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.09673","kind":"arxiv","version":3},"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"}