[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84546-en":3,"doc-seo-84546-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84546,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","A Complete Intersection Theorem for Large Permutation Groups","A family of permutations is called τ-intersecting if any two permutations agree on at least τ elements. The result establishes an integer τ0 such that for every n>τ0 and 1≤τ≤n, the maximum size of a τ-intersecting family inside S_n is achieved by a specific family F_{n,τ,2} defined through the size of the overlap between fixed points and the initial segment {1,…,τ+2}. This provides a large-group analogue of the classical Complete Intersection Theorem and yields an essentially complete solution to the Deza–Frankl intersection problem for permutations.","arXiv :2607 .00318v1 [math .CO] 1 Jul 2026  \nA COMPLETE INTERSECTION THEOREM FOR LARGE PERMUTATION GROUPS  \nNATHAN KELLER, ANDREY KUPAVSKII, NOAM LIFSHITZ, AND OHAD SHEINFELD  \nAbstract . A family of permutations is called 􀁴-intersecting if any two permutations in the family agree on at least 􀁴 elements. We prove that there exists 􀁮0 ∈ N such that for any 􀁮 > 􀁮0 and any 1 ≤ 􀁴 ≤ 􀁮, the maximum size of a 􀁴-intersecting family in 􀁓􀁮 is obtained by one of the families ℱ􀁮,􀁴,􀁲 = {􀀛 ∈ 􀁓􀁮 : |Fixed(􀀛)∩{1 , 2, . . . , 􀁴+2􀁲}| ≥ 􀁴+􀁲}, where Fixed(􀀛) is theset of fixed points of 􀀛 . This proves an analogue of the classical Complete Intersection Theorem for large permutation groups, thus providing an essentially complete solution of the Deza-Frankl intersection problem for permutations (1977) .  \n1. Introduction  \n1.1. Background. A family 􀁆 of subsets of [􀁮] = {1, 2 ,...,􀁮} is 􀁴-intersecting if for any 􀁁,􀁂 ∈ 􀁆, we have |􀁁 ∩ 􀁂| ≥ 􀁴 . For 􀁴 = 1, such families are simply called‘intersecting’. In 1961, Erd˝os, Ko, and Rado [19] proved that for 􀁮 ≥ 􀁮 0 (􀁫,􀁴), the maximum size of a 􀁴-intersecting family of 􀁫-element subsets of [􀁮] is (︀􀁮􀁫−􀁴􀁴)︀ , and asked, what is the minimal number 􀁮0 (􀁫,􀁴) for which this upper bound holds. For 􀁴 = 1, they provided a complete solution, proving that the maximum size is (︀􀁮􀁫−11)︀ for all 􀁫 \u003C ~~􀁮~~2 . This result was highly influential, and by now grew into a subfield of extremal combinatorics, studying collections of objects with forbidden intersections (see the survey [26]) .  \nNaturally, one of the central problems in this field is determining the maximum size of a 􀁴-intersecting family 􀁆 ⊂ 􀁕 , for various ‘universes’ 􀁕 . This problem was studied, e.g., for vector spaces [24], graphs [13], set partitions [35, 38], simplicial complexes [4, 33], linear maps [17], etc. Arguably, the two most thoroughly studied‘universes’ are the original setting where 􀁕 consists of all 􀁫-subsets of [􀁮], and the setting of 􀁴-intersecting families of permutations, i.e., families 􀁆 ⊂ 􀁓􀁮 such that for any 􀀛,􀀜 ∈ 􀁆, there exist 􀁩 1 , . . . ,􀁩 􀁴 with 􀀛 (􀁩􀁪 ) = 􀀜 (􀁩􀁪) for 􀁪 = 1 , . . . ,􀁴 .  \nFor 􀁫-subsets of [􀁮], Frankl [20] determined the minimal 􀁮 0 (􀁫,􀁴) for which the maximum size is (︀􀁮􀁫−􀁴􀁴)︀ for all 􀁴 ≥ 15, and then Wilson [42] determined it for all 􀁴: they showed that 􀁮0 (􀁫,􀁴) = (􀁫 − 􀁴 + 1)(􀁴 + 1) . Furthermore, for all 􀁮 > 􀁮 0 (􀁫,􀁴),  \nDepartment of Mathematics, [Bar-Ilan University.](Bar-Ilan University. Nathan.Keller@biu.ac.il. Supported)[ Nathan.Keller@biu.ac.il](Bar-Ilan University. Nathan.Keller@biu.ac.il. Supported)[. Supported](Bar-Ilan University. Nathan.Keller@biu.ac.il. Supported) by the Israel Science Foundation (grants no. 2669/21 and 2456/25) and by the Binational US-Israel Science Foundation (grant no. 2024120) .  \nMoscow Institute of Physics and Technology, Saint-Petersburg State University, Innopolis Uni[versity.](versity. kupavskii@ya.ru)[ kupavskii@ya.ru](versity. kupavskii@ya.ru).  \nEinstein Institute of Mathematics, [Hebrew University.](Hebrew University. noamlifshitz@gmail.com. Supported)[ noamlifshitz@gmail.com](Hebrew University. noamlifshitz@gmail.com. Supported)[. Supported](Hebrew University. noamlifshitz@gmail.com. Supported)[ ](Hebrew University. noamlifshitz@gmail.com. Supported)by the European Research Council (StG no. 101163794), by the Israel Science Foundation (grant no. 1980/22), and by the Binational US-Israel Science Foundation (grant no. 2024120) .  \nEinstein Institute of Mathematics, [Hebrew University.](Hebrew University. oshenfeld@gmail.com)[ oshenfeld@gmail.com](Hebrew University. oshenfeld@gmail.com).  \nA COMPLETE INTERSECTION THEOREM FOR LARGE PERMUTATION GROUPS 2  \nthe maximum size is obtained only for 􀁴-umvirates, i.e., families of the form {􀁓 ⊂(︀ [􀁮􀁫])︀ : 􀁔 ⊂ 􀁓}, for some |􀁔| = 􀁴 . For the general question of determining the maximum possible size of a 􀁴-intersecting family for any triple (􀁮,􀁫,􀁴), Frankl [20] introduced the families ℱ􀁮,􀁫,􀁴,􀁲 = {􀁓 ⊂ (︀ [􀁮􀁫])︀ : |􀁓 ∩ [􀁴+2􀁲]| ≥ 􀁴+􀁲} and co","cbCaigQfI42yw3RB","https://ap.wps.com/l/cbCaigQfI42yw3RB","pdf",961458,2,1,69,"English","en",105,"# Introduction\n## Background","[{\"question\":\"What does it mean for a family of permutations to be τ-intersecting?\",\"answer\":\"A family is τ-intersecting if any two permutations in the family agree on at least τ elements.\"},{\"question\":\"What family is identified as achieving the maximum size for large n?\",\"answer\":\"For sufficiently large n and for 1≤τ≤n, the maximum τ-intersecting family in S_n is attained by F_{n,τ,2}, characterized via fixed points satisfying a threshold overlap with {1,…,τ+2}.\"},{\"question\":\"Which earlier problem does this theorem address for permutations?\",\"answer\":\"It provides an analogue of the Complete Intersection Theorem for large permutation groups and gives an essentially complete solution to the Deza–Frankl intersection problem for permutations.\"}]",1784196572,174,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-complete-intersection-theorem-for-large-permutation-groups","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-complete-intersection-theorem-for-large-permutation-groups/84546/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-22","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does it mean for a family of permutations to be τ-intersecting?","Question",{"text":75,"@type":76},"A family is τ-intersecting if any two permutations in the family agree on at least τ elements.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What family is identified as achieving the maximum size for large n?",{"text":80,"@type":76},"For sufficiently large n and for 1≤τ≤n, the maximum τ-intersecting family in S_n is attained by F_{n,τ,2}, characterized via fixed points satisfying a threshold overlap with {1,…,τ+2}.",{"name":82,"@type":73,"acceptedAnswer":83},"Which earlier problem does this theorem address for permutations?",{"text":84,"@type":76},"It provides an analogue of the Complete Intersection Theorem for large permutation groups and gives an essentially complete solution to the Deza–Frankl intersection problem for 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