{"paper":{"title":"Topological Ramsey numbers and countable ordinals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Andr\\'es Eduardo Caicedo, Jacob Hilton","submitted_at":"2015-09-30T23:44:33Z","abstract_excerpt":"We study the topological version of the partition calculus in the setting of countable ordinals. Let $\\alpha$ and $\\beta$ be ordinals and let $k$ be a positive integer. We write $\\beta\\to_{top}(\\alpha,k)^2$ to mean that, for every red-blue coloring of the collection of 2-sized subsets of $\\beta$, there is either a red-homogeneous set homeomorphic to $\\alpha$ or a blue-homogeneous set of size $k$. The least such $\\beta$ is the topological Ramsey number $R^{top}(\\alpha,k)$.\n  We prove a topological version of the Erd\\H{o}s-Milner theorem, namely that $R^{top}(\\alpha,k)$ is countable whenever $\\a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.00078","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"}