{"paper":{"title":"Minimal weight in union-closed families","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Victor Falgas--Ravry","submitted_at":"2011-01-13T15:41:05Z","abstract_excerpt":"Let Omega be a finite set and let S be a set system on Omega. For x in Omega, we denote by d_{S}(x) the number of members of S containing x. A long-standing conjecture of Frankl states that if S is union-closed then d(x) \\geq |S|/2 for some x in Omega. We consider a related question. Define the weight of S to be w(S)= \\sum_{A in S} |A|. Suppose S is union-closed. How small can w(S) be? Reimer showed that w(S) \\geq |S| \\log_{2} |S| /2, and that this inequality is sharp. In this paper we show how his bound may be improved if we have some additional information about the domain Omega of S: if S s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.2589","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"}