{"paper":{"title":"Prime Certificates for Exact Vertex-Coprime Ramsey Numbers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM","math.NT"],"primary_cat":"math.CO","authors_text":"Lan Ma, Wenji Xi, Zhicheng Du, Zhuo Deng","submitted_at":"2026-05-26T10:31:20Z","abstract_excerpt":"Let $G_n$ be the coprime graph on $\\{1,\\ldots,n\\}$. We prove that the mixed vertex-coloring coprime Ramsey number satisfies \\[\n  \\Rcop(k_1,\\ldots,k_c)=p_{\\sum_{i=1}^c(k_i-1)}, \\] where $p_m$ is the $m$-th prime. The proof is elementary: the prime clique $\\{1\\}\\cup\\{p\\le n:p\\text{ prime}\\}$ gives the upper bound by pigeonhole, while a prime-bin partition gives the matching lower bound by coloring each composite with a bin containing one of its prime divisors. We reserve $\\Rcop$ for this vertex-coloring parameter; the edge-coloring parameter on the same host graph is denoted $\\Redge$. The same c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.26815","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.26815/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}