{"id":"https://openalex.org/W1531022672","doi":"https://doi.org/10.37236/4548","title":"Shattering-Extremal Set Systems of VC Dimension at most 2","display_name":"Shattering-Extremal Set Systems of VC Dimension at most 2","publication_year":2014,"publication_date":"2014-11-06","ids":{"openalex":"https://openalex.org/W1531022672","doi":"https://doi.org/10.37236/4548","mag":"1531022672"},"language":"en","primary_location":{"id":"doi:10.37236/4548","is_oa":true,"landing_page_url":"https://doi.org/10.37236/4548","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v21i4p30/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/v21i4p30/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5062870828","display_name":"Tam\u00e1s M\u00e9sz\u00e1ros","orcid":"https://orcid.org/0000-0002-4121-7092"},"institutions":[{"id":"https://openalex.org/I198945027","display_name":"Central European University","ror":"https://ror.org/02zx40v98","country_code":"AT","type":"education","lineage":["https://openalex.org/I198945027"]}],"countries":["AT"],"is_corresponding":true,"raw_author_name":"Tam\u00e1s M\u00e9sz\u00e1ros","raw_affiliation_strings":["Central European University, Budapest","Central European University-Budapest"],"affiliations":[{"raw_affiliation_string":"Central European University, Budapest","institution_ids":["https://openalex.org/I198945027"]},{"raw_affiliation_string":"Central European University-Budapest","institution_ids":["https://openalex.org/I198945027"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5034769205","display_name":"Lajos R\u00f3nyai","orcid":"https://orcid.org/0000-0002-8364-9627"},"institutions":[{"id":"https://openalex.org/I7597260","display_name":"Hungarian Academy of Sciences","ror":"https://ror.org/02ks8qq67","country_code":"HU","type":"funder","lineage":["https://openalex.org/I7597260"]},{"id":"https://openalex.org/I4210117195","display_name":"Institute for Computer Science and Control","ror":"https://ror.org/0249v7n71","country_code":"HU","type":"facility","lineage":["https://openalex.org/I4210117195","https://openalex.org/I7597260"]},{"id":"https://openalex.org/I29770179","display_name":"Budapest University of Technology and Economics","ror":"https://ror.org/02w42ss30","country_code":"HU","type":"education","lineage":["https://openalex.org/I29770179"]}],"countries":["HU"],"is_corresponding":false,"raw_author_name":"Lajos R\u00f3nyai","raw_affiliation_strings":["Computer and Automation Research Institute, Hungarian Academy of Sciences and Institute of Mathematics, Budapest University of Technology and Economics","Computer and Automation Research Institute, Hungarian Academy of Sciences and Institute of Mathematics, Budapest University of Technology and Economics#TAB#"],"affiliations":[{"raw_affiliation_string":"Computer and Automation Research Institute, Hungarian Academy of Sciences and Institute of Mathematics, Budapest University of Technology and Economics","institution_ids":["https://openalex.org/I7597260","https://openalex.org/I29770179","https://openalex.org/I4210117195"]},{"raw_affiliation_string":"Computer and Automation Research Institute, Hungarian Academy of Sciences and Institute of Mathematics, Budapest University of Technology and Economics#TAB#","institution_ids":["https://openalex.org/I7597260","https://openalex.org/I29770179","https://openalex.org/I4210117195"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5062870828"],"corresponding_institution_ids":["https://openalex.org/I198945027"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.0400682,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"21","issue":"4","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":0.9994000196456909,"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":0.9994000196456909,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9894000291824341,"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"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9864000082015991,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/corollary","display_name":"Corollary","score":0.8378894329071045},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.802944540977478},{"id":"https://openalex.org/keywords/vc-dimension","display_name":"VC dimension","score":0.777897834777832},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6616278886795044},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6611180305480957},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.6609609723091125},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.40975627303123474},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.12319380044937134}],"concepts":[{"id":"https://openalex.org/C2780012671","wikidata":"https://www.wikidata.org/wiki/Q1343870","display_name":"Corollary","level":2,"score":0.8378894329071045},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.802944540977478},{"id":"https://openalex.org/C119322782","wikidata":"https://www.wikidata.org/wiki/Q2662236","display_name":"VC dimension","level":2,"score":0.777897834777832},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6616278886795044},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6611180305480957},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.6609609723091125},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.40975627303123474},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.12319380044937134},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.37236/4548","is_oa":true,"landing_page_url":"https://doi.org/10.37236/4548","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v21i4p30/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:arXiv.org:1407.3230","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1407.3230","pdf_url":"https://arxiv.org/pdf/1407.3230","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:1531022672","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1407.3230","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":"doi:10.48550/arxiv.1407.3230","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1407.3230","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:10.37236/4548","is_oa":true,"landing_page_url":"https://doi.org/10.37236/4548","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v21i4p30/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":[{"score":0.8199999928474426,"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities"}],"awards":[{"id":"https://openalex.org/G3496479202","display_name":null,"funder_award_id":"NK10564","funder_id":"https://openalex.org/F4320321994","funder_display_name":"Hungarian Scientific Research Fund"},{"id":"https://openalex.org/G6427210305","display_name":null,"funder_award_id":"105645","funder_id":"https://openalex.org/F4320321994","funder_display_name":"Hungarian Scientific Research Fund"},{"id":"https://openalex.org/G7324173974","display_name":null,"funder_award_id":"NK105645","funder_id":"https://openalex.org/F4320321994","funder_display_name":"Hungarian Scientific Research Fund"}],"funders":[{"id":"https://openalex.org/F4320321994","display_name":"Hungarian Scientific Research Fund","ror":"https://ror.org/00v349e63"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1531022672.pdf","grobid_xml":"https://content.openalex.org/works/W1531022672.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W1546490581","https://openalex.org/W1971361630","https://openalex.org/W2020913456","https://openalex.org/W2054603212"],"related_works":["https://openalex.org/W2963693894","https://openalex.org/W2091556712","https://openalex.org/W2131581266","https://openalex.org/W2400081095","https://openalex.org/W2001743814","https://openalex.org/W3105889707","https://openalex.org/W2239848833","https://openalex.org/W2188962096","https://openalex.org/W2766396921","https://openalex.org/W1498760226","https://openalex.org/W2015329281","https://openalex.org/W1995208920","https://openalex.org/W2081255021","https://openalex.org/W2087004834","https://openalex.org/W2072410849","https://openalex.org/W2951236367","https://openalex.org/W2090437514","https://openalex.org/W204859252","https://openalex.org/W2308117615","https://openalex.org/W3151833102"],"abstract_inverted_index":{"We":[0],"say":[1],"that":[2,21],"a":[3,9,24,73],"set":[4,11,25,42,59,86],"system":[5,26,43],"$\\mathcal{F}\\subseteq":[6],"2^{[n]}$":[7],"shatters":[8,28,49],"given":[10],"$S\\subseteq":[12],"[n]$":[13],"if":[14,47],"$2^S=\\{F~\\cap~S":[15],"~:~F~\\in~\\mathcal{F}\\}$.":[16],"The":[17],"Sauer":[18],"inequality":[19],"states":[20],"in":[22,65,88],"general,":[23],"$\\mathcal{F}$":[27],"at":[29],"least":[30],"$|\\mathcal{F}|$":[31,51],"sets.":[32,52],"Here":[33],"we":[34,56,75],"concentrate":[35],"on":[36],"the":[37,89],"case":[38,90],"of":[39,61,67,91,93],"equality.":[40],"A":[41],"is":[44],"called":[45],"shattering-extremal":[46,58,85],"it":[48],"exactly":[50],"In":[53],"this":[54],"paper":[55],"characterize":[57],"systems":[60,87],"Vapnik-Chervonenkis":[62,94],"dimension":[63,95],"$2$":[64],"terms":[66],"their":[68],"inclusion":[69],"graphs,":[70],"and":[71],"as":[72],"corollary":[74],"answer":[76],"an":[77],"open":[78],"question":[79],"about":[80],"leaving":[81],"out":[82],"elements":[83],"from":[84],"families":[92],"$2$.":[96]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":3},{"year":2020,"cited_by_count":2}],"updated_date":"2026-04-10T15:06:20.359241","created_date":"2025-10-10T00:00:00"}