{"id":"https://openalex.org/W2886371064","doi":"https://doi.org/10.37236/8288","title":"VC Dimension and a Union Theorem for Set Systems","display_name":"VC Dimension and a Union Theorem for Set Systems","publication_year":2019,"publication_date":"2019-08-02","ids":{"openalex":"https://openalex.org/W2886371064","doi":"https://doi.org/10.37236/8288","mag":"2886371064"},"language":"en","primary_location":{"id":"doi:10.37236/8288","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8288","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v26i3p24/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v26i3p24/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5015029543","display_name":"Stijn Cambie","orcid":"https://orcid.org/0000-0002-2385-1137"},"institutions":[{"id":"https://openalex.org/I145872427","display_name":"Radboud University Nijmegen","ror":"https://ror.org/016xsfp80","country_code":"NL","type":"education","lineage":["https://openalex.org/I145872427"]}],"countries":["NL"],"is_corresponding":true,"raw_author_name":"Stijn Cambie","raw_affiliation_strings":["Radboud University Nijmegen"],"affiliations":[{"raw_affiliation_string":"Radboud University Nijmegen","institution_ids":["https://openalex.org/I145872427"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5070979870","display_name":"Ant\u00f3nio Gir\u00e3o","orcid":"https://orcid.org/0000-0003-3065-6636"},"institutions":[{"id":"https://openalex.org/I241749","display_name":"University of Cambridge","ror":"https://ror.org/013meh722","country_code":"GB","type":"education","lineage":["https://openalex.org/I241749"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"Ant\u00f3nio Gir\u00e3o","raw_affiliation_strings":["University of Cambridge"],"affiliations":[{"raw_affiliation_string":"University of Cambridge","institution_ids":["https://openalex.org/I241749"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019911064","display_name":"Ross J. Kang","orcid":"https://orcid.org/0000-0002-4219-593X"},"institutions":[{"id":"https://openalex.org/I145872427","display_name":"Radboud University Nijmegen","ror":"https://ror.org/016xsfp80","country_code":"NL","type":"education","lineage":["https://openalex.org/I145872427"]}],"countries":["NL"],"is_corresponding":false,"raw_author_name":"Ross J. Kang","raw_affiliation_strings":["Radboud University Nijmegen"],"affiliations":[{"raw_affiliation_string":"Radboud University Nijmegen","institution_ids":["https://openalex.org/I145872427"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5015029543"],"corresponding_institution_ids":["https://openalex.org/I145872427"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.07282019,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"26","issue":"3","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9993000030517578,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9954000115394592,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10374","display_name":"Advanced Graph Theory Research","score":0.989799976348877,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.786166787147522},{"id":"https://openalex.org/keywords/intersection","display_name":"Intersection (aeronautics)","score":0.7391163110733032},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7372936010360718},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6570135951042175},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.6414173245429993},{"id":"https://openalex.org/keywords/vc-dimension","display_name":"VC dimension","score":0.5814753174781799},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.5009417533874512},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.44190162420272827},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.19284722208976746},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.06900393962860107}],"concepts":[{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.786166787147522},{"id":"https://openalex.org/C64543145","wikidata":"https://www.wikidata.org/wiki/Q162942","display_name":"Intersection (aeronautics)","level":2,"score":0.7391163110733032},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7372936010360718},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6570135951042175},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.6414173245429993},{"id":"https://openalex.org/C119322782","wikidata":"https://www.wikidata.org/wiki/Q2662236","display_name":"VC dimension","level":2,"score":0.5814753174781799},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.5009417533874512},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.44190162420272827},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.19284722208976746},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.06900393962860107},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C146978453","wikidata":"https://www.wikidata.org/wiki/Q3798668","display_name":"Aerospace engineering","level":1,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.37236/8288","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8288","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v26i3p24/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:lirias2repo.kuleuven.be:20.500.12942/738163","is_oa":true,"landing_page_url":"https://lirias.kuleuven.be/handle/20.500.12942/738163","pdf_url":null,"source":{"id":"https://openalex.org/S4306401954","display_name":"Lirias (KU Leuven)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I99464096","host_organization_name":"KU Leuven","host_organization_lineage":["https://openalex.org/I99464096"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"Electronic Journal Of Combinatorics, vol. 26 (3), Art.No. ARTN P3.24","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:arXiv.org:1808.02352","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1808.02352","pdf_url":"https://arxiv.org/pdf/1808.02352","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"text"},{"id":"mag:2886371064","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1808.02352.pdf","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"arXiv (Cornell University)","raw_type":null},{"id":"pmh:ru:oai:repository.ubn.ru.nl:2066/205963","is_oa":true,"landing_page_url":"http://hdl.handle.net/2066/205963","pdf_url":null,"source":{"id":"https://openalex.org/S4306401843","display_name":"Data Archiving and Networked Services (DANS)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I1322597698","host_organization_name":"Royal Netherlands Academy of Arts and Sciences","host_organization_lineage":["https://openalex.org/I1322597698"],"host_organization_lineage_names":[],"type":"repository"},"license":"other-oa","license_id":"https://openalex.org/licenses/other-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"The Electronic Journal of Combinatorics, 26, 1 - 8","raw_type":"info:eu-repo/semantics/article"},{"id":"doi:10.48550/arxiv.1808.02352","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1808.02352","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.37236/8288","is_oa":true,"landing_page_url":"https://doi.org/10.37236/8288","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v26i3p24/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G5365867299","display_name":null,"funder_award_id":"Grant","funder_id":"https://openalex.org/F4320321800","funder_display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek"},{"id":"https://openalex.org/G629491556","display_name":null,"funder_award_id":"(NWO)","funder_id":"https://openalex.org/F4320321800","funder_display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek"},{"id":"https://openalex.org/G8688676036","display_name":null,"funder_award_id":"639.032.614","funder_id":"https://openalex.org/F4320321800","funder_display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek"}],"funders":[{"id":"https://openalex.org/F4320321800","display_name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","ror":"https://ror.org/04jsz6e67"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2886371064.pdf","grobid_xml":"https://content.openalex.org/works/W2886371064.grobid-xml"},"referenced_works_count":8,"referenced_works":["https://openalex.org/W1969201707","https://openalex.org/W1971361630","https://openalex.org/W1981531393","https://openalex.org/W2019935404","https://openalex.org/W2020913456","https://openalex.org/W2047370658","https://openalex.org/W2054603212","https://openalex.org/W2516050763"],"related_works":["https://openalex.org/W2994748271","https://openalex.org/W2229686259","https://openalex.org/W3117099148","https://openalex.org/W2953065566","https://openalex.org/W2952744686","https://openalex.org/W1833887337","https://openalex.org/W2972456824","https://openalex.org/W2912741917","https://openalex.org/W2995687742","https://openalex.org/W2171189166","https://openalex.org/W2000069550","https://openalex.org/W1997726502","https://openalex.org/W2470666513","https://openalex.org/W3094054056","https://openalex.org/W3173369968","https://openalex.org/W2531504305","https://openalex.org/W2963283436","https://openalex.org/W2111427182","https://openalex.org/W2951516747","https://openalex.org/W2950922715"],"abstract_inverted_index":{"Fix":[0],"positive":[1],"integers":[2],"$k$":[3],"and":[4,57,59,96,117,142],"$d$.":[5],"We":[6,98],"show":[7],"that,":[8,70],"as":[9],"$n\\to\\infty$,":[10],"any":[11],"set":[12,82],"system":[13],"$\\mathcal{A}":[14,106],"\\subset":[15,107],"2^{[n]}$":[16,108],"for":[17],"which":[18],"the":[19,43,75,111],"VC":[20,112],"dimension":[21,113],"of":[22,52,55,63,103,114,118,140],"$\\{":[23],"\\triangle_{i=1}^k":[24],"S_i":[25,27],"\\mid":[26],"\\in":[28],"\\mathcal{A}\\}$":[29],"is":[30,48,69,77],"at":[31,36,123],"most":[32,37,124],"$d$":[33],"has":[34],"size":[35],"$(2^{d\\bmod{k}}+o(1))\\binom{n}{\\lfloor":[38],"d/k\\rfloor}$.":[39],"Here":[40],"$\\triangle$":[41],"denotes":[42],"symmetric":[44],"difference":[45],"operator.":[46],"This":[47,132],"a":[49,53,72,104,134],"$k$-fold":[50],"generalisation":[51],"result":[54],"Dvir":[56,141],"Moran,":[58],"it":[60],"settles":[61],"one":[62],"their":[64],"questions.":[65],"A":[66],"key":[67],"insight":[68],"by":[71],"compression":[73],"method,":[74],"problem":[76,84],"equivalent":[78],"to":[79,94,137],"an":[80,101],"extremal":[81],"theoretic":[83],"on":[85],"$k$-wise":[86],"intersection":[87],"or":[88],"union":[89],"that":[90,110],"was":[91],"originally":[92],"due":[93],"Erd\u0151s":[95],"Frankl.":[97],"also":[99],"give":[100],"example":[102],"family":[105],"such":[109],"$\\mathcal{A}\\cap":[115],"\\mathcal{A}$":[116,120],"$\\mathcal{A}\\cup":[119],"are":[121],"both":[122],"$d$,":[125],"while":[126],"$\\lvert":[127],"\\mathcal{A}":[128],"\\rvert":[129],"=":[130],"\\Omega(n^d)$.":[131],"provides":[133],"negative":[135],"answer":[136],"another":[138],"question":[139],"Moran.":[143]},"counts_by_year":[],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}
