{"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"}