[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85940-en":3,"doc-seo-85940-105":29,"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85940,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","A Colorful Extension of VC-dimension and Geometric Applications","VC-dimension measures the complexity of a set system. This paper introduces and studies a colorful variant that models set systems restricted to colored ground sets. The work yields multiple geometric consequences: a strengthened Tverberg theorem for separable abstract convexity spaces with an improved Tverberg number bound, the first colorful k-wise Tverberg theorem in this setting, and quantitative selection, weak epsilon-net, and (p,q)-theorem results. The approach is further extended to colorful Tverberg theorems for unions of convex sets, generalizing earlier uncolored results.","arXiv :2607 . 10496v1 [math .CO] 11 Jul 2026  \nA Colorful Extension of VC-dimension and Geometric Applications  \nChaya Keller∗ and Shakhar Smorodinsky⋆  \nAbstract  \nThe VC-dimension is a fundamental measure of the complexity of a set system. In this paper, we introduce and study a colorful variant of VC-dimension that captures the behavior of set systems on colored ground sets.  \nBy studying this new notion, we obtain a variety of geometric results. First, we prove that separable abstract convexity spaces with Radon number D admit a Tverberg theorem with Tverberg number O (D2 r log r) . This bound significantly improves the O (Dr2 log r) bound of Alon and Smorodinsky from SODA’26 and is the first quasi-linear bound in r, in which the dependence on D is not super-exponential. Second, we prove the first colorful k-wise Tverberg theorem for separable abstract convexity spaces. Using this theorem, we obtain a colorful selection lemma with O (D3 ) colors, an uncolored selection lemma for subsets of size O (D3 ) , a weak ε-net theorem with nets of size OD (ε−O(D3 )), and a (p, q)-theorem with exponent of poly (D) . All these quantitative bounds are significantly better than the best previously known general bounds for abstract convexity spaces. Finally, we extend our method to obtain a colorful Tverberg theorem for unions of convex sets, generalizing the uncolored theorem of Alon and Smorodinsky (SODA’26) .  \n1 Introduction  \nThe VC-dimension is a fundamental measure of the complexity of a set system, with applications in combinatorics, discrete geometry, sampling, and learning theory. In its classical form, the VC-dimension is inherently uncolored: it measures which subsets of a ground set can be realized as traces of hyperedges, without taking into account additional structure such as colors, groups, etc. Various natural combinatorial and geometric settings do carry such extra structure, and the relevant configurations are often rainbow configurations rather than arbitrary subsets. This raisesa basic question: what is the right VC-type notion for colored set systems, and what structural consequences follow from bounded ordinary VC-dimension in such settings?  \nIn this paper we develop a colorful VC framework for this purpose. We introduce k-wise colorful and rainbow shattering parameters for hypergraphs and prove a general upper bound for these parameters in terms of the ordinary VC-dimension. This can be viewed as a colorful analogue of the Perles–Sauer–Shelah lemma: bounded VC-dimension limits the number of rainbow assignments that can be forced across color classes.  \nWe apply this framework to separable abstract convexity spaces. An abstract convexity space is a pair (X, C), where X is a nonempty set and C is a family of subsets of X such that ∅, X ∈ Cand the family C is closed under intersections and under nested unions: if A ⊆ C is nonempty and  \n∗ School of Computer Science, Ariel University, [Israel.](Israel. chayak@ariel.ac.il)[ chayak@ariel.ac.il](Israel. chayak@ariel.ac.il).  \n⋆ Department of Computer Science, Ben-Gurion University of the NEGEV, Be’er Sheva 84105, Israel. [shakhar@bgu.ac.il](shakhar@bgu.ac.il)  \ntotally ordered by inclusion, then SA∈A A ∈ C.1 The sets in C are called convex sets. For a subset A ⊆ X, its convex hull is defined by conv (A) = T {C ∈ C : A ⊆ C} . Thus, conv (A) is the smallest convex set containing A.  \nA subset H ⊆ X is called a halfspace if H ∈ C and X \\ H ∈ C. We say that the convexity space (X, C) is separable if for every two disjoint convex sets A, B ∈ C, there is a halfspace H such that A ⊆ H and B ⊆ X \\ H, and for every convex set C ∈ C and every point x  C, there is a halfspace H such that C ⊆ H and x  H.2  \nFor r ≥ 2, the r-th Tverberg number of (X, C), denoted by Tv C(r), is the smallest integer N, if it exists, such that every set P ⊆ X of size N admits a partition P = P1 ∪˙ · · · ∪˙Pr with Tri=1 conv (Pi)  ∅ . The case r = 2 is the Radon number of the space. Throughout the pape","cbCaibnTLhNjTB1a","https://ap.wps.com/l/cbCaibnTLhNjTB1a","pdf",533215,1,22,"English","en",105,"# Abstract\n# Introduction\n## VC dimension and shattering\n## r-shattering and Tverberg-type background","[{\"question\":\"What is the colorful extension of VC-dimension proposed in the paper?\",\"answer\":\"The paper defines a colorful variant of VC-dimension designed to capture how set systems behave on colored ground sets, focusing on rainbow/color-class structure rather than ordinary traces.\"},{\"question\":\"What major geometric result improves the Tverberg theorem bound?\",\"answer\":\"For separable abstract convexity spaces with Radon number D, the paper proves a Tverberg theorem with a Tverberg number bound that improves the previously known dependence on D and yields a quasi-linear dependence on the number of parts r.\"},{\"question\":\"What additional theorems are derived for colorful settings beyond Tverberg?\",\"answer\":\"Using the colorful k-wise Tverberg theorem, the paper obtains a colorful selection lemma, an uncolored selection lemma for subset sizes, a weak epsilon-net theorem, and a (p,q)-theorem, each with improved quantitative bounds; it also extends to colorful Tverberg theorems for unions of convex sets.\"}]",1784207279,55,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"a-colorful-extension-of-vc-dimension-and-geometric-applications","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/a-colorful-extension-of-vc-dimension-and-geometric-applications/85940/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 is the colorful extension of VC-dimension proposed in the paper?","Question",{"text":75,"@type":76},"The paper defines a colorful variant of VC-dimension designed to capture how set systems behave on colored ground sets, focusing on rainbow/color-class structure rather than ordinary traces.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What major geometric result improves the Tverberg theorem bound?",{"text":80,"@type":76},"For separable abstract convexity spaces with Radon number D, the paper proves a Tverberg theorem with a Tverberg number bound that improves the previously known dependence on D and yields a quasi-linear dependence on the number of parts r.",{"name":82,"@type":73,"acceptedAnswer":83},"What additional theorems are derived for colorful settings beyond Tverberg?",{"text":84,"@type":76},"Using the colorful k-wise Tverberg theorem, the paper obtains a colorful selection lemma, an uncolored selection lemma for subset sizes, a weak epsilon-net theorem, and a (p,q)-theorem, each with improved quantitative bounds; 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