{"paper":{"title":"The 2-color Rado Number of $x_1+x_2+\\cdots +x_{m-1}=ax_m,$ II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dan Saracino","submitted_at":"2013-06-04T13:14:48Z","abstract_excerpt":"In the first installment of this series, we proved that, for every integer $a\\geq 3$ and every $m\\geq 2a^2-a+2$, the 2-color Rado number of $x_1 + x_2 + \\cdots + x_{m-1} = ax_m$ is $\\lceil\\frac{m-1}{a} \\lceil\\frac{m-1}{a} \\rceil\\rceil$. Here we obtain the best possible improvement of the bound on $m.$ We prove that if $3|a$ then the 2-color Rado number is $\\lceil\\frac{m-1}{a} \\lceil\\frac{m-1}{a} \\rceil\\rceil$ when $m\\geq 2a+1$ but not when $m=2a,$ and that if $3\\nmid a$ then the 2-color Rado number is $\\lceil\\frac{m-1}{a} \\lceil\\frac{m-1}{a} \\rceil\\rceil$ when $m\\geq 2a+2$ but not when $m=2a+1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0775","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"}