{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/local-configurations-in-union-closed-families","title":"Local Configurations in Union-Closed Families","arxiv_id":"2301.01331","date":"2023-01-03","proceeding":null,"authors":["Jonad Pulaj","Kenan Wood"],"abstract":"The Frankl or Union-Closed Sets conjecture states that for any finite union-closed family of sets $\\mathcal{F}$ containing some nonempty set, there is some element $i$ in the ground set $U(\\mathcal F) := \\bigcup_{S \\in \\mathcal{F}} S$ of $\\mathcal{F}$ such that $i$ is in at least half of the sets in $\\mathcal{F}$. In this work, we find new values and bounds for the least integer $FC(k, n)$ such that any union-closed family containing $FC(k, n)$ distinct $k$-sets of an $n$-set $X$ satisfies Frankl's conjecture with an element of $X$. Additionally, we answer an older question of Vaughan regarding symmetry in union-closed families and we give a proof of a recent question posed by Ellis, Ivan and Leader. Finally, we introduce novel local configuration criteria through a generalization of Poonen's Theorem to prove the conjecture for many, previously unknown classes of families.","url_abs":"https://arxiv.org/abs/2301.01331v2","url_pdf":"https://arxiv.org/pdf/2301.01331v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"local-configurations-in-union-closed-families","repo_url":"https://github.com/KenanWood/Local-Configurations-in-Union-Closed-Families","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}