{"paper":{"title":"On determination of Zero-sum $\\ell$-generalized Schur Numbers for some linear equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bidisha Roy, Subha Sarkar","submitted_at":"2018-08-27T08:09:44Z","abstract_excerpt":"Let $r$, $m$ and $k\\geq 2$ be positive integers such that $r\\mid k$ and let $v \\in \\left[ 0,\\lfloor \\frac{k-1}{2r} \\rfloor \\right]$ be any integer. For any integer $\\ell \\in [1, k]$ and $\\epsilon \\in \\{0,1\\}$, we let $\\mathcal{E}_{v}^{(\\ell, \\epsilon)}$ be the linear homogeneous equation defined by $\\mathcal{E}_{v}^{(\\ell, \\epsilon)}: x_1 + \\cdots + x_{k-(rv+\\epsilon)} =x_{k-(rv+\\epsilon-1)} +\\cdots+ \\ell x_{k}$. We denote the number $S_{\\mathfrak{z},m}^{(\\ell, \\epsilon)}(k;r;v)$, which is defined to be the least positive integer $t$ such that for any $m$-coloring $\\chi: [1, t] \\to \\{0, 1,\\ldo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.08725","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"}