{"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/further-analysis-on-the-second-frequency-of","title":"Further analysis on the second frequency of union-closed set families","arxiv_id":"2412.03863","date":"2024-12-05","proceeding":null,"authors":["Saintan Wu"],"abstract":"The Union-Closed Sets Conjecture, also known as Frankl's conjecture, asks whether, for any union-closed set family $\\mathcal{F}$ with $m$ sets, there is an element that lies in at least $\\frac{1}{2}\\cdot m$ sets in $\\mathcal{F}$. In 2022, Nagel posed a stronger conjecture that within any union-closed family whose ground set size is at least $k$, there are always $k$ elements in the ground set that appear in at least $\\frac{1}{2^{k-1}+1}$ proportion of the sets in the family. Das and Wu showed that this conjecture is true for $k\\geq 3$ and $k=2$ if $|\\mathcal{F}|$ is outside a particular range. In this companion paper, we analyse further when $\\mathcal{F}$ fails Nagel's conjecture for $k=2$ via linear programming.","url_abs":"https://arxiv.org/abs/2412.03863v2","url_pdf":"https://arxiv.org/pdf/2412.03863v2.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":"further-analysis-on-the-second-frequency-of","repo_url":"https://github.com/haur576/k-union-closed-sets-conjecture","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}