{"id":"https://openalex.org/W2056655718","doi":"https://doi.org/10.1137/140977138","title":"Tur\u00e1n Problems and Shadows III: Expansions of Graphs","display_name":"Tur\u00e1n Problems and Shadows III: Expansions of Graphs","publication_year":2015,"publication_date":"2015-01-01","ids":{"openalex":"https://openalex.org/W2056655718","doi":"https://doi.org/10.1137/140977138","mag":"2056655718"},"language":"en","primary_location":{"id":"doi:10.1137/140977138","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140977138","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://figshare.com/articles/journal_contribution/TURAN_PROBLEMS_AND_SHADOWS_III_EXPANSIONS_OF_GRAPHS/10773011","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5080894223","display_name":"Alexandr Kostochka","orcid":"https://orcid.org/0000-0002-6363-3804"},"institutions":[{"id":"https://openalex.org/I157725225","display_name":"University of Illinois Urbana-Champaign","ror":"https://ror.org/047426m28","country_code":"US","type":"education","lineage":["https://openalex.org/I157725225"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Alexandr Kostochka","raw_affiliation_strings":["University of Illinois at Urbana Champaign"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Illinois at Urbana Champaign","institution_ids":["https://openalex.org/I157725225"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079803922","display_name":"Dhruv Mubayi","orcid":"https://orcid.org/0000-0002-4709-8768"},"institutions":[{"id":"https://openalex.org/I39422238","display_name":"University of Illinois Chicago","ror":"https://ror.org/02mpq6x41","country_code":"US","type":"education","lineage":["https://openalex.org/I39422238"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Dhruv Mubayi","raw_affiliation_strings":["University of Illinois at Chicago"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of Illinois at Chicago","institution_ids":["https://openalex.org/I39422238"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5024446850","display_name":"Jacques Verstra\u00ebte","orcid":null},"institutions":[{"id":"https://openalex.org/I36258959","display_name":"University of California San Diego","ror":"https://ror.org/0168r3w48","country_code":"US","type":"education","lineage":["https://openalex.org/I36258959"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jacques Verstra\u00ebte","raw_affiliation_strings":["University of California at San Diego"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"University of California at San Diego","institution_ids":["https://openalex.org/I36258959"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":4.4442,"has_fulltext":false,"cited_by_count":27,"citation_normalized_percentile":{"value":0.94334975,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"29","issue":"2","first_page":"868","last_page":"876"},"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":0.9998999834060669,"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":0.9998999834060669,"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/T10374","display_name":"Advanced Graph Theory Research","score":0.9940999746322632,"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/T11476","display_name":"Graph theory and applications","score":0.9930999875068665,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/combinatorics","display_name":"Combinatorics","score":0.9047646522521973},{"id":"https://openalex.org/keywords/hypergraph","display_name":"Hypergraph","score":0.8304464817047119},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.779025673866272},{"id":"https://openalex.org/keywords/bipartite-graph","display_name":"Bipartite graph","score":0.5880909562110901},{"id":"https://openalex.org/keywords/disjoint-sets","display_name":"Disjoint sets","score":0.5646403431892395},{"id":"https://openalex.org/keywords/vertex","display_name":"Vertex (graph theory)","score":0.5205217599868774},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.4533977508544922},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4297563135623932}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.9047646522521973},{"id":"https://openalex.org/C2781221856","wikidata":"https://www.wikidata.org/wiki/Q840247","display_name":"Hypergraph","level":2,"score":0.8304464817047119},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.779025673866272},{"id":"https://openalex.org/C197657726","wikidata":"https://www.wikidata.org/wiki/Q174733","display_name":"Bipartite graph","level":3,"score":0.5880909562110901},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.5646403431892395},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.5205217599868774},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.4533977508544922},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4297563135623932}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/140977138","is_oa":false,"landing_page_url":"https://doi.org/10.1137/140977138","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.768.5903","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.768.5903","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://arxiv.org/pdf/1407.2588.pdf","raw_type":"text"},{"id":"pmh:oai:figshare.com:article/10773011","is_oa":true,"landing_page_url":"https://figshare.com/articles/journal_contribution/TURAN_PROBLEMS_AND_SHADOWS_III_EXPANSIONS_OF_GRAPHS/10773011","pdf_url":null,"source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"}],"best_oa_location":{"id":"pmh:oai:figshare.com:article/10773011","is_oa":true,"landing_page_url":"https://figshare.com/articles/journal_contribution/TURAN_PROBLEMS_AND_SHADOWS_III_EXPANSIONS_OF_GRAPHS/10773011","pdf_url":null,"source":{"id":"https://openalex.org/S4377196282","display_name":"Figshare","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210132348","host_organization_name":"Figshare (United Kingdom)","host_organization_lineage":["https://openalex.org/I4210132348"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2074406313","display_name":"Extremal Combinatorics and Applications","funder_award_id":"1362650","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G28412016","display_name":null,"funder_award_id":"15\u201001\u201005867","funder_id":"https://openalex.org/F4320321079","funder_display_name":"Russian Foundation for Basic Research"},{"id":"https://openalex.org/G3252936920","display_name":"Extremal and Probabilistic Questions on Hypergraphs","funder_award_id":"1300138","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G3960888090","display_name":null,"funder_award_id":"DMS-1266016","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4305763183","display_name":"Extremal and Probabilistic questions on hypergraphs","funder_award_id":"0969092","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G4596003035","display_name":null,"funder_award_id":"12-01-00631","funder_id":"https://openalex.org/F4320321079","funder_display_name":"Russian Foundation for Basic Research"},{"id":"https://openalex.org/G4708938160","display_name":null,"funder_award_id":"DMS\u20101300138","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G6079940770","display_name":null,"funder_award_id":"DMS-1362650","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8562604245","display_name":null,"funder_award_id":"DMS-0969092","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320321079","display_name":"Russian Foundation for Basic Research","ror":"https://ror.org/02mh1ke95"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W37455515","https://openalex.org/W1444168583","https://openalex.org/W1582742149","https://openalex.org/W1648902078","https://openalex.org/W1964094911","https://openalex.org/W1977374427","https://openalex.org/W1998787536","https://openalex.org/W2009852526","https://openalex.org/W2019935404","https://openalex.org/W2029485128","https://openalex.org/W2048996648","https://openalex.org/W2059283278","https://openalex.org/W2090526201","https://openalex.org/W2100896331","https://openalex.org/W2111430057","https://openalex.org/W2148318379","https://openalex.org/W2154774722","https://openalex.org/W2282353092","https://openalex.org/W2950710487","https://openalex.org/W2951699198"],"related_works":["https://openalex.org/W4376608589","https://openalex.org/W3138003926","https://openalex.org/W4300037846","https://openalex.org/W1630514295","https://openalex.org/W1537073411","https://openalex.org/W2963081352","https://openalex.org/W4376608938","https://openalex.org/W4288275998","https://openalex.org/W2472555608","https://openalex.org/W4214498971"],"abstract_inverted_index":{"The":[0,60],"expansion":[1],"$G^+$":[2],"of":[3,18,42,55,62,74,101,104,126],"a":[4,21,45,56,80,84,91,98],"graph":[5,155],"$G$":[6,13,19,78,82,89,137,151,175],"is":[7,79,83,90,131,141,152,176],"the":[8,39,69,102,127,138,177],"3-uniform":[9,46,57],"hypergraph":[10,47,58],"obtained":[11],"from":[12,25],"by":[14,33],"enlarging":[15],"each":[16],"edge":[17],"with":[20,48,156],"new":[22],"vertex":[23],"disjoint":[24,76],"$V(G)$":[26],"such":[27],"that":[28,71,147],"distinct":[29,34],"edges":[30,43],"are":[31],"enlarged":[32],"vertices.":[35],"Let":[36],"${ex}_3(n,F)$":[37],"denote":[38],"maximum":[40],"number":[41,158],"in":[44,143,167],"$n$":[49],"vertices":[50],"not":[51],"containing":[52],"any":[53,153],"copy":[54],"$F$.":[59],"study":[61,100],"${ex}_3(n,G^+)$":[63,140],"includes":[64],"some":[65],"well-researched":[66],"problems,":[67],"including":[68],"case":[70],"$F$":[72],"consists":[73],"$k$":[75],"edges,":[77],"triangle,":[81],"path":[85],"or":[86],"cycle,":[87],"and":[88,121,166],"tree.":[92],"In":[93],"this":[94,148,169],"paper":[95],"we":[96,107],"initiate":[97],"broader":[99],"behavior":[103],"${ex}_3(n,G^+)$.":[105],"Specifically,":[106],"show":[108,146],"$":[109],"{ex}_3(n,K_{s,t}^+)":[110],"=":[111,162,172],"\\Theta(n^{3":[112],"-":[113,119],"3/s})$":[114],"whenever":[115],"$t":[116],">":[117],"(s":[118],"1)!$":[120],"$s":[122],"\\geq":[123],"3$.":[124],"One":[125],"main":[128],"open":[129],"problems":[130],"to":[132],"determine":[133],"for":[134],"which":[135],"graphs":[136],"quantity":[139],"quadratic":[142],"$n$.":[144],"We":[145],"occurs":[149],"when":[150,174],"bipartite":[154],"Tur\u00e1n":[157],"$o(n^{\\varphi})$":[159],"where":[160],"$\\varphi":[161],"\\frac{1":[163],"+":[164],"\\sqrt{5}}{2}$,":[165],"particular":[168],"shows":[170],"${ex}_3(n,G^+)":[171],"O(n^2)$":[173],"three-dimensional":[178],"cube":[179],"graph.":[180]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":1},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":2},{"year":2014,"cited_by_count":1}],"updated_date":"2026-08-01T09:00:35.917206","created_date":"2025-10-10T00:00:00"}
