{"paper":{"title":"Zero-sum Analogues of van der Waerden's Theorem on Arithmetic Progressions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aaron Robertson","submitted_at":"2018-02-09T18:50:03Z","abstract_excerpt":"Let $r$ and $k$ be positive integers with $r \\mid k$. Denote by $w_{\\mathrm{\\mathfrak{z}}}(k;r)$ the minimum integer such that every coloring $\\chi:[1,w_{\\mathrm{\\mathfrak{z}}}(k;r)] \\rightarrow \\{0,1,\\dots,r-1\\}$ admits a $k$-term arithmetic progression $a,a+d,\\dots,a+(k-1)d$ with $\\sum_{j=0}^{k-1} \\chi(a+jd) \\equiv 0 \\,(\\mathrm{mod }\\,r)$. We investigate these numbers as well as a \"mixed\" monochromatic/zero-sum analogue. We also present an interesting reciprocity between the van der Waerden numbers and $w_{\\mathrm{\\mathfrak{z}}}(k;r)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.03387","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"}