{"id":"https://openalex.org/W4408477355","doi":"https://doi.org/10.1007/s00373-025-02909-6","title":"VC-Dimension of Hyperplanes Over Finite Fields","display_name":"VC-Dimension of Hyperplanes Over Finite Fields","publication_year":2025,"publication_date":"2025-03-15","ids":{"openalex":"https://openalex.org/W4408477355","doi":"https://doi.org/10.1007/s00373-025-02909-6"},"language":"en","primary_location":{"id":"doi:10.1007/s00373-025-02909-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00373-025-02909-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00373-025-02909-6.pdf","source":{"id":"https://openalex.org/S186126824","display_name":"Graphs and Combinatorics","issn_l":"0911-0119","issn":["0911-0119","1435-5914"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Graphs and Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007/s00373-025-02909-6.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5062336525","display_name":"Ruben Ascoli","orcid":"https://orcid.org/0009-0007-2906-0807"},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Ruben Ascoli","raw_affiliation_strings":["Department of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA"],"raw_orcid":"https://orcid.org/0009-0007-2906-0807","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA","institution_ids":["https://openalex.org/I130701444"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5072566696","display_name":"Livia Betti","orcid":null},"institutions":[{"id":"https://openalex.org/I5388228","display_name":"University of Rochester","ror":"https://ror.org/022kthw22","country_code":"US","type":"education","lineage":["https://openalex.org/I5388228"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Livia Betti","raw_affiliation_strings":["Department of Mathematics, University of Rochester, Rochester, NY, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Rochester, Rochester, NY, USA","institution_ids":["https://openalex.org/I5388228"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5086175220","display_name":"Justin Cheigh","orcid":null},"institutions":[{"id":"https://openalex.org/I22675022","display_name":"Williams College","ror":"https://ror.org/04avkmd49","country_code":"US","type":"education","lineage":["https://openalex.org/I22675022"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Justin Cheigh","raw_affiliation_strings":["Department of Mathematics, Williams College, Williamstown, MA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Williams College, Williamstown, MA, USA","institution_ids":["https://openalex.org/I22675022"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5071497714","display_name":"Alex Iosevich","orcid":"https://orcid.org/0000-0001-9074-9864"},"institutions":[{"id":"https://openalex.org/I5388228","display_name":"University of Rochester","ror":"https://ror.org/022kthw22","country_code":"US","type":"education","lineage":["https://openalex.org/I5388228"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Alex Iosevich","raw_affiliation_strings":["Department of Mathematics, University of Rochester, Rochester, NY, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Rochester, Rochester, NY, USA","institution_ids":["https://openalex.org/I5388228"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031430372","display_name":"Ryan Jeong","orcid":null},"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":"Ryan Jeong","raw_affiliation_strings":["Department of Mathematics, University of Cambridge, Cambridge, England"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Cambridge, Cambridge, England","institution_ids":["https://openalex.org/I241749"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5039669454","display_name":"Xuyan Liu","orcid":"https://orcid.org/0000-0003-3379-6888"},"institutions":[{"id":"https://openalex.org/I27804330","display_name":"Brown University","ror":"https://ror.org/05gq02987","country_code":"US","type":"education","lineage":["https://openalex.org/I27804330"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Xuyan Liu","raw_affiliation_strings":["Department of Mathematics, Brown University, Providence, RI, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Brown University, Providence, RI, USA","institution_ids":["https://openalex.org/I27804330"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5056023044","display_name":"Brian McDonald","orcid":"https://orcid.org/0000-0001-8600-5096"},"institutions":[{"id":"https://openalex.org/I165733156","display_name":"University of Georgia","ror":"https://ror.org/00te3t702","country_code":"US","type":"education","lineage":["https://openalex.org/I165733156"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Brian McDonald","raw_affiliation_strings":["Department of Mathematics, University of Georgia, Athens, GA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Georgia, Athens, GA, USA","institution_ids":["https://openalex.org/I165733156"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5057348386","display_name":"Wyatt Milgrim","orcid":null},"institutions":[{"id":"https://openalex.org/I161318765","display_name":"University of California, Los Angeles","ror":"https://ror.org/046rm7j60","country_code":"US","type":"education","lineage":["https://openalex.org/I161318765"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Wyatt Milgrim","raw_affiliation_strings":["Department of Mathematics, University of California Los Angeles, Los Angeles, CA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of California Los Angeles, Los Angeles, CA, USA","institution_ids":["https://openalex.org/I161318765"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5049983178","display_name":"Steven J. Miller","orcid":"https://orcid.org/0000-0003-4904-4605"},"institutions":[{"id":"https://openalex.org/I22675022","display_name":"Williams College","ror":"https://ror.org/04avkmd49","country_code":"US","type":"education","lineage":["https://openalex.org/I22675022"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Steven J. Miller","raw_affiliation_strings":["Department of Mathematics, Williams College, Williamstown, MA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Williams College, Williamstown, MA, USA","institution_ids":["https://openalex.org/I22675022"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5050786693","display_name":"Francisco Romero Acosta","orcid":null},"institutions":[{"id":"https://openalex.org/I859038795","display_name":"Virginia Tech","ror":"https://ror.org/02smfhw86","country_code":"US","type":"education","lineage":["https://openalex.org/I859038795"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Francisco Romero Acosta","raw_affiliation_strings":["Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA","institution_ids":["https://openalex.org/I859038795"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5026477624","display_name":"Santiago Velazquez Iannuzzelli","orcid":null},"institutions":[{"id":"https://openalex.org/I111979921","display_name":"Northwestern University","ror":"https://ror.org/000e0be47","country_code":"US","type":"education","lineage":["https://openalex.org/I111979921"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Santiago Velazquez Iannuzzelli","raw_affiliation_strings":["Department of Mathematics, Northwestern University, Evanston, IL, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Northwestern University, Evanston, IL, USA","institution_ids":["https://openalex.org/I111979921"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":11,"corresponding_author_ids":["https://openalex.org/A5062336525"],"corresponding_institution_ids":["https://openalex.org/I130701444"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.03614769,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"41","issue":"2","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9988999962806702,"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"}},"topics":[{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":0.9988999962806702,"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/T10849","display_name":"Finite Group Theory Research","score":0.9973000288009644,"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/T12504","display_name":"Mathematics and Applications","score":0.9965000152587891,"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/hyperplane","display_name":"Hyperplane","score":0.897253155708313},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8447957038879395},{"id":"https://openalex.org/keywords/finite-field","display_name":"Finite field","score":0.670685350894928},{"id":"https://openalex.org/keywords/dimension","display_name":"Dimension (graph theory)","score":0.6596575975418091},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.631300687789917},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.39122557640075684}],"concepts":[{"id":"https://openalex.org/C68693459","wikidata":"https://www.wikidata.org/wiki/Q657586","display_name":"Hyperplane","level":2,"score":0.897253155708313},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8447957038879395},{"id":"https://openalex.org/C77926391","wikidata":"https://www.wikidata.org/wiki/Q603880","display_name":"Finite field","level":2,"score":0.670685350894928},{"id":"https://openalex.org/C33676613","wikidata":"https://www.wikidata.org/wiki/Q13415176","display_name":"Dimension (graph theory)","level":2,"score":0.6596575975418091},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.631300687789917},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.39122557640075684}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s00373-025-02909-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00373-025-02909-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00373-025-02909-6.pdf","source":{"id":"https://openalex.org/S186126824","display_name":"Graphs and Combinatorics","issn_l":"0911-0119","issn":["0911-0119","1435-5914"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Graphs and Combinatorics","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1007/s00373-025-02909-6","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s00373-025-02909-6","pdf_url":"https://link.springer.com/content/pdf/10.1007/s00373-025-02909-6.pdf","source":{"id":"https://openalex.org/S186126824","display_name":"Graphs and Combinatorics","issn_l":"0911-0119","issn":["0911-0119","1435-5914"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Graphs and Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G7315378982","display_name":null,"funder_award_id":"DMS2154232","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"}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4408477355.pdf"},"referenced_works_count":20,"referenced_works":["https://openalex.org/W1573648177","https://openalex.org/W1758887938","https://openalex.org/W2040033904","https://openalex.org/W2045383484","https://openalex.org/W2101788557","https://openalex.org/W2167283930","https://openalex.org/W2269710820","https://openalex.org/W2343027471","https://openalex.org/W2582801014","https://openalex.org/W2953161972","https://openalex.org/W2964335955","https://openalex.org/W3205023862","https://openalex.org/W4210453869","https://openalex.org/W4236362309","https://openalex.org/W4238284510","https://openalex.org/W4293763988","https://openalex.org/W4387502508","https://openalex.org/W4392135020","https://openalex.org/W4401639904","https://openalex.org/W4407092331"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2369655046","https://openalex.org/W2019467317","https://openalex.org/W1585519779","https://openalex.org/W1965515989","https://openalex.org/W2088737283","https://openalex.org/W1999587863","https://openalex.org/W1929902734","https://openalex.org/W4287205200","https://openalex.org/W2964302506"],"abstract_inverted_index":{"Abstract":[0],"Let":[1],"$$\\mathbb":[2],"{F}_q^d$$":[3,30,252],"<mml:math":[4,31,50,67,98,128,157,169,199,233,253,269],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\">":[5,32,51,68,99,129,158,170,200,234,254,270],"<mml:msubsup>":[6,36,258],"<mml:mi>F</mml:mi>":[7,37,56,259],"<mml:mi>q</mml:mi>":[8,38,57,180,260,283],"<mml:mi>d</mml:mi>":[9,39,160,261,280,285,290],"</mml:msubsup>":[10,40,262],"</mml:math>":[11,42,60,93,116,143,164,187,212,246,264,298],"be":[12],"the":[13,19,118,122,147,154,194,220,228],"d":[14,248],"-dimensional":[15],"vector":[16],"space":[17],"over":[18],"finite":[20],"field":[21],"with":[22,149,265],"q":[23,189],"elements.":[24],"For":[25],"a":[26,44],"subset":[27],"$$E\\subseteq":[28,250],"\\mathbb":[29,48,251],"<mml:mrow>":[33,52,69,74,80,100,108,130,159,171,172,201,206,235,240,255,271,272,284,289],"<mml:mi>E</mml:mi>":[34,76,89,106,134,174,208,242,256,274],"<mml:mo>\u2286</mml:mo>":[35,257],"</mml:mrow>":[41,59,78,91,92,114,115,142,163,176,186,210,211,244,245,263,276,293,295,297],"and":[43,188],"fixed":[45],"nonzero":[46],"$$t\\in":[47],"{F}_q$$":[49],"<mml:mi>t</mml:mi>":[53,72,140,204,238],"<mml:mo>\u2208</mml:mo>":[54,88,133],"<mml:msub>":[55,70,82,101,202,236,278],"</mml:msub>":[58,73,85,104,205,239,281],",":[61,94],"let":[62],"$$\\mathcal":[63,197,231],"{H}_t(E)=\\{h_y:":[64],"y\\in":[65],"E\\}$$":[66],"<mml:mi>H</mml:mi>":[71,203,237],"<mml:mo>(</mml:mo>":[75,207,241],"<mml:mo>)</mml:mo>":[77,209,243],"<mml:mo>=</mml:mo>":[79,139,161],"<mml:mo>{</mml:mo>":[81,109,131],"<mml:mi>h</mml:mi>":[83,102],"<mml:mi>y</mml:mi>":[84,87,103,138],"<mml:mo>:</mml:mo>":[86,105,135],"<mml:mo>}</mml:mo>":[90,113,141],"where":[95],"$$h_y:E\\rightarrow":[96],"\\{0,1\\}$$":[97],"<mml:mo>\u2192</mml:mo>":[107],"<mml:mn>0</mml:mn>":[110],"<mml:mo>,</mml:mo>":[111],"<mml:mn>1</mml:mn>":[112,288,292],"is":[117,190,213,247],"indicator":[119],"function":[120],"of":[121,146,196,230],"set":[123],"$$\\{x\\in":[124],"E:":[125],"x\\cdot":[126],"y=t\\}$$":[127],"<mml:mi>x</mml:mi>":[132,136],"<mml:mo>\u00b7</mml:mo>":[137],".":[144,299],"Two":[145],"authors,":[148],"Maxwell":[150],"Sun,":[151],"showed":[152],"in":[153],"case":[155],"$$d=3$$":[156],"<mml:mn>3</mml:mn>":[162],"that":[165,227],"if":[166],"$$|E|\\ge":[167,266],"Cq^{\\frac{11}{4}}$$":[168],"<mml:mo>|</mml:mo>":[173,175,273,275],"<mml:mo>\u2265</mml:mo>":[177,277],"<mml:mi>C</mml:mi>":[178,279],"<mml:msup>":[179,282],"<mml:mfrac>":[181,287],"<mml:mn>11</mml:mn>":[182],"<mml:mn>4</mml:mn>":[183],"</mml:mfrac>":[184,294],"</mml:msup>":[185,296],"sufficiently":[191],"large,":[192],"then":[193],"VC-dimension":[195,229],"{H}_t(E)$$":[198,232],"3.":[214],"In":[215],"this":[216],"paper,":[217],"we":[218],"generalize":[219],"result":[221],"to":[222],"arbitrary":[223],"dimension":[224],"by":[225],"showing":[226],"whenever":[249],"C_d":[267],"q^{d-\\frac{1}{d-1}}$$":[268],"<mml:mo>-</mml:mo>":[286,291]},"counts_by_year":[],"updated_date":"2025-12-21T01:58:51.020947","created_date":"2025-10-10T00:00:00"}