[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-267275-105":59,"doc-detail-267275-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","almost-all-permutation-matrices-have-bounded-saturation-functions-proof","Almost All Permutation Matrices Have Bounded Saturation Functions - Proof","","Saturation problems for forbidden graphs are studied through forbidden 0-1 matrices and associated saturation functions. A 0-1 matrix is saturating for a forbidden matrix P if it avoids P while changing any zero to one forces a copy of P. For sat(n;P), a key dichotomy is that sat(n;P) is either bounded by a constant or grows on the order of n. The paper proves that almost all k×k permutation matrices have bounded saturation functions and develops a characterization approach beyond previously known examples.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/almost-all-permutation-matrices-have-bounded-saturation-functions-proof/267275/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/almost-all-permutation-matrices-have-bounded-saturation-functions-proof/267275.png","ImageObject",300,407,{"name":92,"@type":93},"Asher","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-27","2026-09-14",true,{"@type":102,"interactionType":103,"userInteractionCount":52},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does it mean for a 0-1 matrix to be saturating for a forbidden matrix P?","Question",{"text":112,"@type":113},"A 0-1 matrix avoids P, but changing any zero entry to one creates a copy of P within the matrix.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is the key dichotomy for sat(n; P) established by Fulek and Keszegh?",{"text":117,"@type":113},"For every forbidden 0-1 matrix P, the saturation function either stays O(1) or grows on the order of n.",{"name":119,"@type":110,"acceptedAnswer":120},"What does the paper prove about permutation matrices P?",{"text":121,"@type":113},"It verifies the conjecture that almost all k×k permutation matrices P have sat(n;P)=O(1), and it constructs a family of permutation matrices exhibiting bounded saturation behavior.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},267275,1789416592,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":52,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":129,"read_time":144,"preview_page_count":4},687197207639,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Almost all permutation matrices have bounded saturation functions  \nJesse Geneson  \nDepartment of Mathematics & Statistics  \nSan Jose State University  \nCalifornia, U.S.A.  \n[jesse.geneson@sjsu.edu](jesse.geneson@sjsu.edu)  \nSubmitted: Dec 28, 2020; Accepted: Apr 17, 2021; Published: May 7, 2021 © The author. Released under the CC BY license (International 4.0) .  \nAbstract  \nSaturation problems for forbidden graphs have been a popular area of research for many decades, and recently Brualdi and Cao initiated the study of a saturation problem for 0-1 matrices. We say that a 0-1 matrix A is saturating for the forbidden 0-1 matrix P if A avoids P but changing any zero to a one in A creates a copy of P. De􀀌ne sat(n; P ) to be the minimum possible number of ones in an n 􀀂 n 0-1 matrix that is saturating for P. Fulek and Keszegh proved that for every 0-1 matrix P , either sat(n; P ) = O(1) or sat(n; P ) = 􀀂(n) . They found two 0-1 matrices P for which sat(n; P ) = O(1), as well as in􀀌nite families of 0-1 matrices P for which sat(n; P ) = 􀀂(n) . Their results imply that sat(n; P ) = 􀀂(n) for almost all k 􀀂 k 0-1 matrices P.  \nFulek and Keszegh conjectured that there are many more 0-1 matrices P such that sat(n; P ) = O(1) besides the ones they found, and they asked for a characterization of all permutation matrices P such that sat(n; P ) = O(1) . We a􀀎rm their conjecture by proving that almost all k 􀀂 k permutation matrices P have sat(n; P ) = O(1) . We also make progress on the characterization problem, since our proof of the main result exhibits a family of permutation matrices with bounded saturation functions.  \nMathematics Subject Classi􀀌cations: 05D99  \n1 Introduction  \nWe say that a 0-1 matrix A contains the 0-1 matrix P if some submatrix of A is either equal to P or can be turned into P by changing some ones to zeroes. Otherwise, we say that A avoids P. The extremal function ex(n; P ) is de􀀌ned as the maximum number of ones in an n 􀀂 n 0-1 matrix that avoids P. This extremal function has been applied to a  \nthe electronic journal of combinatorics 28(2) (2021), \\#P2.16 [https://doi.org/10.37236/10124](https://doi.org/10.37236/10124)  \nwide range of problems including the proof of the Stanley-Wilf conjecture [33], the best known bounds on the maximum number of unit distances in a convex n-gon [16], a bound on the complexity of an algorithm for 􀀌nding a minimal path in a rectlilinear grid with obstacles [35], and bounds on extremal functions of forbidden sequences [38] .  \nThere is a long history of the study of extremal functions of forbidden 0-1 matrices, see e.g. [40 , 32 , 2 , 37 , 26 , 6 , 24] . Marcus and Tardos [33] proved that ex(n; P ) = O(n) for every permutation matrix P , and used this fact to resolve the F􀁿uredi-Hajnal conjecture [17], which also resolved the Stanley-Wilf conjecture using an earlier result of Klazar [29] . In a di􀀋erent paper, Tardos 􀀌nished bounding the extremal functions of all forbidden 0-1 matrices with at most four ones up to a constant factor [41] . Many connections have been found between extremal functions of forbidden 0-1 matrices and the maximum possible lengths of generalized Davenport-Schinzel sequences [5 , 19 , 21 , 36 , 39 , 42] .  \nBesides permutation matrices, other families of forbidden 0-1 matrices with linear extremal functions have been found including 0-1 matrices obtained by doubling permutation matrices [20], 0-1 matrices corresponding to visibility graphs [11 , 22], and 0-1 matrices obtained by performing operations on smaller 0-1 matrices with linear extremal functions [27 , 38] . After the result of Marcus and Tardos, Fox [10] proved that almost all k 􀀂 k permutation matrices have ex(n; P ) = 2􀀊( pk)n. Fox also showed that ex(n; P ) = 2O (k)n for every k 􀀂 k permutation matrix P , and these results for permutation matrices were also generalized to extremal functions of forbidden d-dimensional permutation matrices [30 , 23 , 4 , 34] .  \nSaturation problems have been studied","cbCaiaOYUAqw8PrA","https://ap.wps.com/l/cbCaiaOYUAqw8PrA","pdf",259520,13,"English","# Introduction\n## Forbidden submatrices and extremal functions\n## Saturation functions and known bounds\n## Conjectures and main result","[{\"question\":\"What does it mean for a 0-1 matrix to be saturating for a forbidden matrix P?\",\"answer\":\"A 0-1 matrix avoids P, but changing any zero entry to one creates a copy of P within the matrix.\"},{\"question\":\"What is the key dichotomy for sat(n; P) established by Fulek and Keszegh?\",\"answer\":\"For every forbidden 0-1 matrix P, the saturation function either stays O(1) or grows on the order of n.\"},{\"question\":\"What does the paper prove about permutation matrices P?\",\"answer\":\"It verifies the conjecture that almost all k×k permutation matrices P have sat(n;P)=O(1), and it constructs a family of permutation matrices exhibiting bounded saturation behavior.\"}]","Almost All Permutation Matrices Have Bounded Saturation Functions - Proof | PDF",33]