{"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/on-dedekind-s-problem-a-sparse-version-of","title":"On Dedekind's problem, a sparse version of Sperner's theorem, and antichains of a given size in the Boolean lattice","arxiv_id":"2411.03400","date":"2024-11-05","proceeding":null,"authors":["Matthew Jenssen","Alexandru Malekshahian","Jinyoung Park"],"abstract":"Dedekind's problem, dating back to 1897, asks for the total number $\\psi(n)$ of antichains contained in the Boolean lattice $B_n$ on $n$ elements. We study Dedekind's problem using a recently developed method based on the cluster expansion from statistical physics and as a result, obtain several new results on the number and typical structure of antichains in $B_n$. We obtain detailed estimates for both $\\psi(n)$ and the number of antichains of size $\\beta \\binom{n}{\\lfloor n/2 \\rfloor}$ for any fixed $\\beta>0$. We also establish a sparse version of Sperner's theorem: we determine the sharp threshold and scaling window for the property that almost every antichain of size $m$ is contained in a middle layer of $B_n$.","url_abs":"https://arxiv.org/abs/2411.03400v2","url_pdf":"https://arxiv.org/pdf/2411.03400v2.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":"on-dedekind-s-problem-a-sparse-version-of","repo_url":"https://github.com/collares/cluster-coefficients","is_official":1,"mentioned_in_paper":0,"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}