[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81890-en":3,"doc-seo-81890-105":31,"detail-sidebar-cat-0-en-105":93},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},81890,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Ample Sets in Cartesian Products","Ample sets of hypercubes, introduced by A. Dress in 1995, are subsets with a shattering-to-strong-shattering property. This work extends the theory to Cartesian products of finite sets U=U1×…×Um by using minor-subproducts built from partitioning and contracting factors. For S⊆U, shattering, copy, projection, and strong-projection are defined on minor-subproducts. A central theorem characterizes ampleness via complement ampleness, isometricity/superisometricity, and commutativity over disjoint supports, with efficient criteria, decomposition results, and applications to VC-dimension in multiclass learning.","arXiv :2607 .04014v1 [math .CO] 4 Jul 2026  \nAMPLE SETS IN CARTESIAN PRODUCTS  \nVICTOR CHEPOI 1 ,3 AND MATTHEW MAAT2  \n1 LIS, Aix-Marseille Université, CNRS, and Université de Toulon  \nFaculté des Sciences de Luminy, F-13288 Marseille Cedex 9, France  \n[victor.chepoi@lis-lab.fr](victor.chepoi@lis-lab.fr)  \n2 University of Twente, The Netherlands  \n[m.t.maat@utwente.nl](m.t.maat@utwente.nl)  \n3 Institut Universitaire de France (IUF)  \nAbstract . Ample sets of hypercubes, introduced by A. Dress in 1995, constitute an interesting combinatorial structure with rich properties and important examples. They are the subsets S of the hypercube {0, 1}E such that any subhypercube {0, 1}Y , Y ⊆ E shattered by S is strongly shattered by S. Ample sets can be characterized in a multitude of combinatorial, graph-theoretical, recursive, and geometrical ways, and they are equivalent to lopsided sets introduced by J. Lawrence in 1983 .  \nIn this paper, we define and investigate ample sets of Cartesian products of finite sets, i.e. of U = U1 × ... × Um . This is done using minor-subproducts of U , which correspond to products of partitions of the factors U1 , . . . , Um: each minor-subproduct M is obtained by partitioning each Ui into blocks and contracting each block into a single element. For a minor-subproduct M and a set S ⊆ U , we define the notions of shattering of M by S , of copy of M in S , of projection SM of S on M , and of strong-projection S M of S on M. We call a set S ⊆ U ample if for any minor-subproduct M that is shattered by S , there exists a copy of M included in S. Using the lattice structure of minor-subproducts, we also define lopsided sets. Differently from the binary case, ampleness is no longer equivalent to lopsidedness.  \nWe prove however that several characterizations of classical ample sets can be extended to ample sets of Cartesian products. In particular, we show that ampleness of S is equivalent to any of the following: ampleness of the complement S ∗ = U \\ S , isometricity of SM for any minor-subproduct M (superisometricity), and commutativity (SM )M ′ = (SM′)M for all minorsubproducts M, M ′ with disjoint supports. We also provide more efficient characterizations of ampleness, in particular, by showing that S is ample if and only of S is isometric and both Seand Se are ample for some elementary minor-subproduct, if and only if the intersection of S with any interval [u, v] with u, v ∈ S is ample in the classical sense. We also characterize ampleness by push downs and provide a decomposition theorem for ample sets, allowing us to prove that the prism complexes of ample sets are contractible. We provide new examples of ample sets arising from payoff games in graphs, prism-like polyhedra, and quasi-median graphs. Finally, we provide a unified treatment of various notions of VC-dimension occurring in the literature on multiclass learning in terms of the dimension of shattered minor-subproducts of certain types.  \n1. Introduction  \n1.1. Avant-propos. Projection is a fundamental mathematical operation. For example, given a set system S ⊆ 2E = {0, 1}E and a subset Y ⊆ E , the trace S |Y = {A ∩ Y : A ∈ S} of Son Y is the projection of S (viewed as a subset of vertices of the hypercube {0, 1}E ) on the hypercube {0, 1}Y . If every element of {0, 1}Y is in the image of S (i.e. , S |Y = 2Y ), then Y (or the hypercube {0, 1}Y ) is said to be shattered by S. The set X(S) of all shattered sets is asimplicial complex (a set system closed by taking subsets) and the dimension of this complex (the largest size of a set of X(S)) is the well-known Vapnik-Chervonenkis dimension ( VC-dimension for short) of S. On the other hand, the hypercube {0, 1}E is partitioned into copies of {0, 1}Y ,  \nDate: July 7, 2026 .  \n2 V. CHEPOI AND M. MAAT  \ni.e. , into “parallel” cubes of the form s × {0, 1}Y with s ∈ {0, 1}E\\Y . Then Y (or 2Y ) is said tobe strongly shattered by S if at least one such copy is included in S. The set X (S) of all strong","cbCaikIBc8SqTt5y","https://ap.wps.com/l/cbCaikIBc8SqTt5y","pdf",963185,6,1,56,"English","en",105,"# Introduction\n## Avant-propos (Projection, shattering, strong shattering)\n## Classical ample/lopsided/extremal/simple sets","[{\"question\":\"What is an ample set in this context?\",\"answer\":\"A set S ⊆ U is ample if for every minor-subproduct M that is shattered by S, S contains a copy of M. This generalizes the shattering-to-strong-shattering principle beyond binary hypercubes.\"},{\"question\":\"How are minor-subproducts of a Cartesian product constructed?\",\"answer\":\"Each minor-subproduct M is obtained by partitioning every factor Ui into blocks and contracting each block into a single element. This yields a corresponding product structure on the contracted elements.\"},{\"question\":\"What equivalent characterizations of ampleness are proved?\",\"answer\":\"Ampleness of S is equivalent to ampleness of the complement S* = U\\\\S, to isometricity/superisometricity of projections on any minor-subproduct M, and to a commutativity identity (SM)M′ = (SM′)M for minors with disjoint supports.\"}]","Ample Sets in Cartesian Products | PDF",1784176890,141,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":88,"head_meta":90,"extra_data":92,"updated_unix":29},"ample-sets-in-cartesian-products","",{"@graph":37,"@context":87},[38,55,70],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":54},"https://docshare.wps.com/document/ample-sets-in-cartesian-products/81890/",4,{"url":53,"name":13,"@type":56,"author":57,"headline":13,"publisher":59,"fileFormat":62,"inLanguage":24,"description":14,"dateModified":63,"datePublished":64,"encodingFormat":62,"isAccessibleForFree":65,"interactionStatistic":66},"DigitalDocument",{"name":9,"@type":58},"Person",{"url":42,"name":60,"@type":61},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":67,"interactionType":68,"userInteractionCount":20},"InteractionCounter",{"@type":69},"ViewAction",{"@type":71,"mainEntity":72},"FAQPage",[73,79,83],{"name":74,"@type":75,"acceptedAnswer":76},"What is an ample set in this context?","Question",{"text":77,"@type":78},"A set S ⊆ U is ample if for every minor-subproduct M that is shattered by S, S contains a copy of M. This generalizes the shattering-to-strong-shattering principle beyond binary hypercubes.","Answer",{"name":80,"@type":75,"acceptedAnswer":81},"How are minor-subproducts of a Cartesian product constructed?",{"text":82,"@type":78},"Each minor-subproduct M is obtained by partitioning every factor Ui into blocks and contracting each block into a single element. This yields a corresponding product structure on the contracted elements.",{"name":84,"@type":75,"acceptedAnswer":85},"What equivalent characterizations of ampleness are proved?",{"text":86,"@type":78},"Ampleness of S is equivalent to ampleness of the complement S* = U\\S, to isometricity/superisometricity of projections on any minor-subproduct M, and to a commutativity identity (SM)M′ = (SM′)M for minors with disjoint supports.","https://schema.org",{"og:url":53,"og:type":89,"og:title":13,"og:site_name":60,"og:description":14},"article",{"robots":91,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":94},[95,99,103,107,112,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":96,"show_sort_weight":97,"slug":98},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":100,"show_sort_weight":101,"slug":102},"Literature",80,"literature",{"id":54,"doc_module":4,"doc_module_name":47,"category_name":104,"show_sort_weight":105,"slug":106},"Exam",70,"exam",{"id":108,"doc_module":4,"doc_module_name":47,"category_name":109,"show_sort_weight":110,"slug":111},5,"Comic",60,"comic",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":108,"slug":139},19,"General","general"]