{"paper":{"title":"Large B_d-free and union-free subfamilies","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bal\\'azs Patk\\'os, Ida Kantor, J\\'anos Bar\\'at, Younjin Kim, Zolt\\'an F\\\"uredi","submitted_at":"2010-12-17T16:33:32Z","abstract_excerpt":"For a property $\\Gamma$ and a family of sets $\\cF$, let $f(\\cF,\\Gamma)$ be the size of the largest subfamily of $\\cF$ having property $\\Gamma$. For a positive integer $m$, let $f(m,\\Gamma)$ be the minimum of $f(\\cF,\\Gamma)$ over all families of size $m$. A family $\\cF$ is said to be $B_d$-free if it has no subfamily $\\cF'=\\{F_I: I \\subseteq [d]\\}$ of $2^d$ distinct sets such that for every $I,J \\subseteq [d]$, both $F_I \\cup F_J=F_{I \\cup J}$ and $F_I \\cap F_J = F_{I \\cap J}$ hold. A family $\\cF$ is $a$-union free if $F_1\\cup ... F_a \\neq F_{a+1}$ whenever $F_1,..,F_{a+1}$ are distinct sets in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3918","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"}