{"paper":{"title":"The 2-color Rado number of $x_1+x_2+\\cdots +x_n=y_1+y_2+\\cdots +y_k$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dan Saracino","submitted_at":"2014-02-24T14:06:38Z","abstract_excerpt":"In 1982, Beutelspacher and Brestovansky determined the 2-color Rado number of the equation $$x_1+x_2+\\cdots +x_{m-1}=x_m$$ for all $m\\geq 3.$ Here we extend their result by determining the 2-color Rado number of the equation $$x_1+x_2+\\cdots +x_n=y_1+y_2+\\cdots +y_k$$ for all $n\\geq 2$ and $k\\geq 2.$ As a consequence, we determine the 2-color Rado number of $$x_1+x_2+\\cdots +x_n=a_1y_1+\\cdots +a_{\\ell}y_{\\ell}$$ in all cases where $n\\geq 2$ and $n\\geq a_1+\\cdots +a_{\\ell},$ and in most cases where $n\\geq 2$ and $2n\\geq a_1+\\cdots +a_{\\ell}.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1402.5829","kind":"arxiv","version":2},"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"}