{"paper":{"title":"Selective and Ramsey ultrafilters on $G$-spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GN"],"primary_cat":"math.CO","authors_text":"I.V. Protasov, O.V. Petrenko","submitted_at":"2013-10-07T15:47:15Z","abstract_excerpt":"Let $G$ be a group, $X$ be an infinite transitive $G$-space. A free ultrafilter $\\UU$ on $X$ is called $G$-selective if, for any $G$-invariant partition $\\PP$ of $X$, either one cell of $\\PP$ is a member of $\\UU$, or there is a member of $\\UU$ which meets each cell of $\\PP$ in at most one point. We show (Theorem 1) that in ZFC with no additional set-theoretical assumptions there exists a $G$-selective ultrafilter on $X$, describe all $G$-spaces $X$ (Theorem 2) such that each free ultrafilter on $X$ is $G$-selective, and prove (Theorem 3) that a free ultrafilter $\\UU$ on $\\omega$ is selective i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.1827","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"}