[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83433-en":3,"doc-seo-83433-105":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},83433,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","An Almost Complete 𝛛-Intersection Theorem for Permutations","For any ε > 0 and n > (1 + ε)k, n > n0(ε), the work determines the size of the largest k-intersecting family of permutations and establishes a sharp stability result. The results resolve a conjecture of Ellis, Friedgut and Pilpel (2011) and confirm the predictions of Frankl and Deza (1977) and Cameron (1988) in the stated parameter range. Extremal families are not necessarily trivial and the statement is analogous to the Ahlswede-Khachatrian theorem, with proof based on refined spread approximations.","arXiv :2405 .07843v1 [math .CO] 13 May 2024  \nAN ALMOST COMPLETE 􀁴-INTERSECTION THEOREM  \nFOR PERMUTATIONS  \nAbstract . For any 􀀏 > 0 and 􀁮 > (1 + 􀀏)􀁴 , 􀁮 > 􀁮0 (􀀏) we determine the size of the largest 􀁴-intersecting family of permutations, as well as give a sharp stability result. This resolves a conjecture of Ellis, Friedgut and Pilpel (2011) and shows the validity of conjectures of Frankl and Deza (1977) and Cameron (1988) for 􀁮 > (1+􀀏)􀁴 . We note that, for this range of parameters, the extremal examples are not necessarily trivial, and that our statement is analogous to the celebrated Ahlswede-Khachatrian theorem. The proof is based on the refinement of the method of spread approximations, recently introduced by Kupavskii and Zakharov (2022) .  \n1. Introduction  \nLet [􀁮] = {1,...,􀁮} stand for the standard 􀁮-element set and let 2[􀁮], (︀ [􀁮􀁫])︀ denote its power set and the set of all 􀁫-element subsets. One of the classical results in extremal combinatorics is the Erd˝os–Ko–Rado theorem [12] . We say that a family of sets is intersecting if any two sets from the family have non-empty intersection. The EKR theorem states that for 􀁮 ⩾ 2􀁫 any intersecting family ℱ ⊂ (︀ [􀁮􀁫])︀ has size at most (︀􀁮􀁫−11)︀ . This result was highly influential, and by now grew into a subfield of extremal combinatorics, studying collections of objects with forbidden intersections.  \nErd˝os, Ko and Rado [12] also showed that the largest 􀁴-intersecting family of sets in (︀ [􀁮􀁫])︀ has size (︀􀁮􀁫−􀁴􀁴)︀ , provided 􀁮 > 􀁮0 (􀁫) . A family is 􀁴-intersecting if any two sets from the family intersect in at least 􀁴 elements. Note that the extremal example for this result, as well as the ‘classical’EKR theorem, is the family of all sets that contain a fixed 􀁴-element set. Later, Frankl [13] and Wilson [38] determined the exact value of 􀁮0 (􀁫): the same conclusion holds for 􀁮 ⩾ (􀁴 + 1)(􀁫 − 􀁴 + 1) . For smaller values of 􀁮, other 􀁴-intersecting families become larger. For 􀁩 = 0 ,...,􀁫 − 􀁴 define the Frankl families  \nℬ 􀁩 = {︁􀁆 ∈ (︂ [􀁮􀁫])︂ : |􀁆 ∩ [􀁴 + 2􀁩]| ⩾ 􀁴 + 􀁩 }︁ .  \nFor 􀁮 \u003C (􀁴 + 1)(􀁫 − 􀁴 + 1) |ℬ1 | > |ℬ0 | . It turned out, however, that for any values of 􀁮,􀁫,􀁴 one of ℬ 􀁩 should be extremal. Frankl [13] introduced the families defined above and conjectured that for every 􀁮,􀁫,􀁴 one of themis extremal. This was shown for a wide range of parameters by Frankl and F¨uredi [18] and then for all 􀁮,􀁫,􀁴 by Ahlswede and Khachatrian [1] in the socalled ‘Complete 􀁴-intersection Theorem’. This theorem played an important role in some applications to computer science, in particular the result of Dinur and Safra [7] from hardness of approximation.  \n2 AN ALMOST COMPLETE 􀁴-INTERSECTION THEOREM FOR PERMUTATIONS  \nAnother very influential forbidden intersections result is due to Frankland Wilson [22] . It addresses the so-called Erd˝os–S´os problem: determine the largest family in (︀ [􀁮􀁫])︀ with intersection exactly 􀁴 forbidden. Importantly, it gives an almost sharp result for the case when 􀁫,􀁴 are linear in 􀁮 (under some number-theoretic restrictions) . This was extremely important for applications, and the Frankl–Wilson theorem and a more general Frankl–R¨odl theorem [20] was used for several questions in discrete geometry and Ramsey theory (for applications in discrete geometry, see [21], [35], [36]) .  \nForbidden intersections were studied for structures other than families in 2 [􀁮] or (︀ [􀁮􀁫])︀ : graphs [8], partitions [34] [29], simplicial complexes [4] [30], vector spaces [19], and permutations. Permutations are by far the most studied object in this respect. Denote by Σ􀁮 the collection of permutations [􀁮] → [􀁮] . The study of forbidden intersection theorems for permutations goes back to the paper of Frankl and Deza [15], in which they studied the question of how big a family of permutations from Σ􀁮 could be, if any two permutations 􀀛1 ,􀀛2 from the family agree on at least 􀁴 points: satisfy 􀀛1 (􀁸) =􀀛2 (􀁸) for at least 􀁴 different 􀁸 ∈ [􀁮] . They showed that for 􀁴 = 1 the answer","cbCaij4bTJZUR7lU","https://ap.wps.com/l/cbCaij4bTJZUR7lU","pdf",532192,2,1,17,"English","en",105,"# Introduction\n## Classical intersection theorems\n## Frankl–Wilson and forbidden intersections\n## Forbidden intersections for permutations\n## Frankl–Deza conjecture and related collections\n## Ellis–Friedgut–Pilpel conjecture","[{\"question\":\"What does the paper determine about k-intersecting families of permutations?\",\"answer\":\"It determines the size of the largest k-intersecting family of permutations under conditions on n relative to k, and it proves a sharp stability result for extremal families.\"},{\"question\":\"Which conjectures are resolved or validated by the results?\",\"answer\":\"The paper resolves a conjecture of Ellis, Friedgut and Pilpel (2011) and shows the validity of conjectures of Frankl and Deza (1977) and Cameron (1988) in the relevant parameter range.\"},{\"question\":\"What is the main idea behind the proof approach?\",\"answer\":\"The proof is based on a refinement of the method of spread approximations, introduced by Kupavskii and Zakharov (2022).\"}]",1784187689,43,{"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},"an-almost-complete-intersection-theorem-for-permutations","",{"@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/an-almost-complete-intersection-theorem-for-permutations/83433/",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","2026-07-24","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 the paper determine about k-intersecting families of permutations?","Question",{"text":75,"@type":76},"It determines the size of the largest k-intersecting family of permutations under conditions on n relative to k, and it proves a sharp stability result for extremal families.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which conjectures are resolved or validated by the results?",{"text":80,"@type":76},"The paper resolves a conjecture of Ellis, Friedgut and Pilpel (2011) and shows the validity of conjectures of Frankl and Deza (1977) and Cameron (1988) in the relevant parameter range.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the main idea behind the proof approach?",{"text":84,"@type":76},"The proof is based on a refinement of the method of spread approximations, introduced by Kupavskii and Zakharov 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