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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1310.1827","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-07T15:47:15Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"3cf4c4ba16aecb67941df5ecab1b4e66b325aecc964dd4b366fb46a7e0a1263b","abstract_canon_sha256":"8aae9f2ba1489001edc132e0e14560705ad6d63036e98ac938cb7afd082f9344"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:45.439673Z","signature_b64":"DrxJe8AV6UJHM53MdSLK7fFOC0jLHxfYJngEkfzzcSN4/dkhH4H+c4gy9tOneCsJP45pUFEXYQoPxBvmcrr0Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb7330820f70d8ddbbe8432b9677720197dfe7d2c3b051cc2213dafddaeee184","last_reissued_at":"2026-05-18T00:32:45.439072Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:45.439072Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1310.1827","created_at":"2026-05-18T00:32:45.439160+00:00"},{"alias_kind":"arxiv_version","alias_value":"1310.1827v1","created_at":"2026-05-18T00:32:45.439160+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.1827","created_at":"2026-05-18T00:32:45.439160+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZNZTBAQPODMN","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZNZTBAQPODMN3O7I","created_at":"2026-05-18T12:28:09.283467+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZNZTBAQP","created_at":"2026-05-18T12:28:09.283467+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZNZTBAQPODMN3O7IIMVZM53SAG","json":"https://pith.science/pith/ZNZTBAQPODMN3O7IIMVZM53SAG.json","graph_json":"https://pith.science/api/pith-number/ZNZTBAQPODMN3O7IIMVZM53SAG/graph.json","events_json":"https://pith.science/api/pith-number/ZNZTBAQPODMN3O7IIMVZM53SAG/events.json","paper":"https://pith.science/paper/ZNZTBAQP"},"agent_actions":{"view_html":"https://pith.science/pith/ZNZTBAQPODMN3O7IIMVZM53SAG","download_json":"https://pith.science/pith/ZNZTBAQPODMN3O7IIMVZM53SAG.json","view_paper":"https://pith.science/paper/ZNZTBAQP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1310.1827&json=true","fetch_graph":"https://pith.science/api/pith-number/ZNZTBAQPODMN3O7IIMVZM53SAG/graph.json","fetch_events":"https://pith.science/api/pith-number/ZNZTBAQPODMN3O7IIMVZM53SAG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZNZTBAQPODMN3O7IIMVZM53SAG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZNZTBAQPODMN3O7IIMVZM53SAG/action/storage_attestation","attest_author":"https://pith.science/pith/ZNZTBAQPODMN3O7IIMVZM53SAG/action/author_attestation","sign_citation":"https://pith.science/pith/ZNZTBAQPODMN3O7IIMVZM53SAG/action/citation_signature","submit_replication":"https://pith.science/pith/ZNZTBAQPODMN3O7IIMVZM53SAG/action/replication_record"}},"created_at":"2026-05-18T00:32:45.439160+00:00","updated_at":"2026-05-18T00:32:45.439160+00:00"}