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We show that if $\\fff$ is $\\ee$-filling over $\\ccc$ and $C$-measurable in $[\\ccc]^{<\\omega}$, then for every $P\\subseteq\\ccc$ perfect there exists $Q\\subseteq P$ perfect with $[Q]^{<\\omega}\\subseteq\\fff$. A similar result for weaker versions of density is also obtained."},"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":"0805.2031","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2008-05-14T11:21:27Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e44a29c3f3fa1a3a30dd46665401cbbf9571031dd8368f18de38ec983dfdb1fa","abstract_canon_sha256":"6ff0311c4f3178f11be51c647477df739da4a24db242c09dcf4af7117684d8b5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:47:21.368233Z","signature_b64":"DozGDBCgi1ACxl39stcVh4yn5rW5adCAxZrODgzWIO6IVXoVNfKW684z+c6rs8pfyxUDA7EXxLa2mWjqchkqDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f82c6b15619a33569093cdaceec1f4249efa3e1d507fd3f545276205c3bee581","last_reissued_at":"2026-05-17T23:47:21.367811Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:47:21.367811Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On filling families of finite subsets of the Cantor set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.LO","authors_text":"Pandelis Dodos, Vassilis Kanellopoulos","submitted_at":"2008-05-14T11:21:27Z","abstract_excerpt":"Let $\\ee>0$ and $\\fff$ be a family of finite subsets of the Cantor set $\\ccc$. Following D. H. Fremlin, we say that $\\fff$ is $\\ee$-filling over $\\ccc$ if $\\fff$ is hereditary and for every $F\\subseteq\\ccc$ finite there exists $G\\subseteq F$ such that $G\\in\\fff$ and $|G|\\geq\\ee |F|$. We show that if $\\fff$ is $\\ee$-filling over $\\ccc$ and $C$-measurable in $[\\ccc]^{<\\omega}$, then for every $P\\subseteq\\ccc$ perfect there exists $Q\\subseteq P$ perfect with $[Q]^{<\\omega}\\subseteq\\fff$. A similar result for weaker versions of density is also obtained."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0805.2031","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":"0805.2031","created_at":"2026-05-17T23:47:21.367867+00:00"},{"alias_kind":"arxiv_version","alias_value":"0805.2031v1","created_at":"2026-05-17T23:47:21.367867+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0805.2031","created_at":"2026-05-17T23:47:21.367867+00:00"},{"alias_kind":"pith_short_12","alias_value":"7AWGWFLBTIZV","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_16","alias_value":"7AWGWFLBTIZVNEET","created_at":"2026-05-18T12:25:56.245647+00:00"},{"alias_kind":"pith_short_8","alias_value":"7AWGWFLB","created_at":"2026-05-18T12:25:56.245647+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/7AWGWFLBTIZVNEETZWWO5QPUES","json":"https://pith.science/pith/7AWGWFLBTIZVNEETZWWO5QPUES.json","graph_json":"https://pith.science/api/pith-number/7AWGWFLBTIZVNEETZWWO5QPUES/graph.json","events_json":"https://pith.science/api/pith-number/7AWGWFLBTIZVNEETZWWO5QPUES/events.json","paper":"https://pith.science/paper/7AWGWFLB"},"agent_actions":{"view_html":"https://pith.science/pith/7AWGWFLBTIZVNEETZWWO5QPUES","download_json":"https://pith.science/pith/7AWGWFLBTIZVNEETZWWO5QPUES.json","view_paper":"https://pith.science/paper/7AWGWFLB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0805.2031&json=true","fetch_graph":"https://pith.science/api/pith-number/7AWGWFLBTIZVNEETZWWO5QPUES/graph.json","fetch_events":"https://pith.science/api/pith-number/7AWGWFLBTIZVNEETZWWO5QPUES/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7AWGWFLBTIZVNEETZWWO5QPUES/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7AWGWFLBTIZVNEETZWWO5QPUES/action/storage_attestation","attest_author":"https://pith.science/pith/7AWGWFLBTIZVNEETZWWO5QPUES/action/author_attestation","sign_citation":"https://pith.science/pith/7AWGWFLBTIZVNEETZWWO5QPUES/action/citation_signature","submit_replication":"https://pith.science/pith/7AWGWFLBTIZVNEETZWWO5QPUES/action/replication_record"}},"created_at":"2026-05-17T23:47:21.367867+00:00","updated_at":"2026-05-17T23:47:21.367867+00:00"}