{"paper":{"title":"Upper and Lower Bounds on Zero-Sum Generalized Schur Numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Erik Metz","submitted_at":"2018-08-11T19:12:08Z","abstract_excerpt":"Let $S_{\\mathfrak{z}}(k,r)$ be the least positive integer such that for any $r$-coloring $\\chi : \\{1,2,\\dots,S_{\\mathfrak{z}}(k,r)\\} \\longrightarrow \\{1, 2, \\dots, r\\}$, there is a sequence $x_1, x_2, \\dots, x_k$ such that $\\sum_{i=1}^{k-1} x_i = x_k$, and $\\sum_{i=1}^{k} \\chi(x_i) \\equiv 0 \\pmod{r}$. We show that when $k$ is greater than $r$, $kr - r - 1 \\le S_{\\mathfrak{z}}(k,r) \\le kr - 1$, and when $r$ is an odd prime, $S_{\\mathfrak{z}}(k,r)$ is in fact equal to $kr - r$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.03851","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"}