{"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/3-setwise-intersecting-families-of-the","title":"$3$-setwise intersecting families of the symmetric group","arxiv_id":"2010.00229","date":"2020-10-01","proceeding":null,"authors":["Angelot Behajaina","Roghayeh Maleki","Aina Toky Rasoamanana","A. Sarobidy Razafimahatratra"],"abstract":"Given two positive integers $n\\geq 3$ and $t\\leq n$, the permutations $\\sigma,\\pi \\in \\operatorname{Sym}(n)$ are $t$-setwise intersecting if they agree (setwise) on a $t$-subset of $\\{1,2,\\ldots,n\\}$. A family $\\mathcal{F} \\subset \\operatorname{Sym}(n)$ is $t$-setwise intersecting if any two permutations of $\\mathcal{F}$ are $t$-setwise intersecting. Ellis [Journal of Combinatorial Theory, Series A, 119(4), 825--849, 2012] conjectured that if $t\\leq n$ and $\\mathcal{F} \\subset \\operatorname{Sym}(n)$ is a $t$-setwise intersecting family, then $|\\mathcal{F}|\\leq t!(n-t)!$ and equality holds only if $\\mathcal{F}$ is a coset of a setwise stablizer of a $t$-subset of $\\{1,2,\\ldots,n\\}$. In this paper, we prove that if $n\\geq 11$ and $\\mathcal{F}$ is $3$-setwise intersecting, then $|\\mathcal{F}|\\leq 6(n-3)!$. Moreover, we prove that the characteristic vector of a $3$-setwise intersecting family of maximum size lies in the sum of the eigenspaces induced by the permutation module of $\\operatorname{Sym}(n)$ acting on the $3$-subsets of $\\{1,2,\\ldots,n\\}$.","url_abs":"https://arxiv.org/abs/2010.00229v2","url_pdf":"https://arxiv.org/pdf/2010.00229v2.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":"3-setwise-intersecting-families-of-the","repo_url":"https://gitlab.com/sarobidy19/3-setwise-intersecting-permutations","is_official":1,"mentioned_in_paper":1,"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}