{"paper":{"title":"A constant lower bound for the union-closed sets conjecture","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Justin Gilmer","submitted_at":"2022-11-16T17:17:42Z","abstract_excerpt":"We show that for any union-closed family $\\mathcal{F} \\subseteq 2^{[n]}, \\mathcal{F} \\neq \\{\\emptyset\\}$, there exists an $i \\in [n]$ which is contained in a $0.01$ fraction of the sets in $\\mathcal{F}$. This is the first known constant lower bound, and improves upon the $\\Omega(\\log_2(|\\mathcal{F}|)^{-1})$ bounds of Knill and W\\'{o}jick. Our result follows from an information theoretic strengthening of the conjecture. Specifically, we show that if $A, B$ are independent samples from a distribution over subsets of $[n]$ such that $Pr[i \\in A] < 0.01$ for all $i$ and $H(A) > 0$, then $H(A \\cup "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.09055","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2211.09055/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}