{"id":"https://openalex.org/W2775536822","doi":"https://doi.org/10.1007/s10623-018-0499-9","title":"Covers and partial transversals of Latin squares","display_name":"Covers and partial transversals of Latin squares","publication_year":2018,"publication_date":"2018-06-20","ids":{"openalex":"https://openalex.org/W2775536822","doi":"https://doi.org/10.1007/s10623-018-0499-9","mag":"2775536822"},"language":"en","primary_location":{"id":"doi:10.1007/s10623-018-0499-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10623-018-0499-9","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs10623-018-0499-9.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["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":"Designs, Codes and Cryptography","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"hybrid","oa_url":"https://link.springer.com/content/pdf/10.1007%2Fs10623-018-0499-9.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5033250524","display_name":"Darcy Best","orcid":null},"institutions":[{"id":"https://openalex.org/I56590836","display_name":"Monash University","ror":"https://ror.org/02bfwt286","country_code":"AU","type":"education","lineage":["https://openalex.org/I56590836"]}],"countries":["AU"],"is_corresponding":true,"raw_author_name":"Darcy Best","raw_affiliation_strings":["School of Mathematics, Monash University, Melbourne, Australia","School of Mathematics Monash University Melbourne, Australia"],"affiliations":[{"raw_affiliation_string":"School of Mathematics, Monash University, Melbourne, Australia","institution_ids":["https://openalex.org/I56590836"]},{"raw_affiliation_string":"School of Mathematics Monash University Melbourne, Australia","institution_ids":["https://openalex.org/I56590836"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5037471767","display_name":"Trent G. Marbach","orcid":"https://orcid.org/0000-0002-3708-3095"},"institutions":[{"id":"https://openalex.org/I56590836","display_name":"Monash University","ror":"https://ror.org/02bfwt286","country_code":"AU","type":"education","lineage":["https://openalex.org/I56590836"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Trent Marbach","raw_affiliation_strings":["School of Mathematics, Monash University, Melbourne, Australia","School of Mathematics Monash University Melbourne, Australia"],"affiliations":[{"raw_affiliation_string":"School of Mathematics, Monash University, Melbourne, Australia","institution_ids":["https://openalex.org/I56590836"]},{"raw_affiliation_string":"School of Mathematics Monash University Melbourne, Australia","institution_ids":["https://openalex.org/I56590836"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5003936493","display_name":"Rebecca J. Stones","orcid":"https://orcid.org/0000-0002-9991-2995"},"institutions":[{"id":"https://openalex.org/I205237279","display_name":"Nankai University","ror":"https://ror.org/01y1kjr75","country_code":"CN","type":"education","lineage":["https://openalex.org/I205237279"]},{"id":"https://openalex.org/I98301712","display_name":"Baidu (China)","ror":"https://ror.org/03vs3wt56","country_code":"CN","type":"company","lineage":["https://openalex.org/I98301712"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Rebecca J. Stones","raw_affiliation_strings":["Nankai-Baidu Joint Laboratory, College of Computer and Control Engineering & College of Software, Nankai University, Tianjin, China"],"affiliations":[{"raw_affiliation_string":"Nankai-Baidu Joint Laboratory, College of Computer and Control Engineering & College of Software, Nankai University, Tianjin, China","institution_ids":["https://openalex.org/I205237279","https://openalex.org/I98301712"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5019627457","display_name":"Ian M. Wanless","orcid":"https://orcid.org/0000-0001-8085-0387"},"institutions":[{"id":"https://openalex.org/I56590836","display_name":"Monash University","ror":"https://ror.org/02bfwt286","country_code":"AU","type":"education","lineage":["https://openalex.org/I56590836"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Ian M. Wanless","raw_affiliation_strings":["School of Mathematics, Monash University, Melbourne, Australia","School of Mathematics Monash University Melbourne, Australia"],"affiliations":[{"raw_affiliation_string":"School of Mathematics, Monash University, Melbourne, Australia","institution_ids":["https://openalex.org/I56590836"]},{"raw_affiliation_string":"School of Mathematics Monash University Melbourne, Australia","institution_ids":["https://openalex.org/I56590836"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5033250524"],"corresponding_institution_ids":["https://openalex.org/I56590836"],"apc_list":{"value":2290,"currency":"EUR","value_usd":2890},"apc_paid":{"value":2290,"currency":"EUR","value_usd":2890},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.00281552,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"87","issue":"5","first_page":"1109","last_page":"1136"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11797","display_name":"graph theory and CDMA systems","score":1.0,"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":1.0,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9969000220298767,"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/T11130","display_name":"Coding theory and cryptography","score":0.9965000152587891,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/transversal","display_name":"Transversal (combinatorics)","score":0.95920729637146},{"id":"https://openalex.org/keywords/latin-square","display_name":"Latin square","score":0.8829907178878784},{"id":"https://openalex.org/keywords/cover","display_name":"Cover (algebra)","score":0.7628360986709595},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7354923486709595},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7242065668106079},{"id":"https://openalex.org/keywords/square","display_name":"Square (algebra)","score":0.6517934799194336},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.627597987651825},{"id":"https://openalex.org/keywords/bar","display_name":"Bar (unit)","score":0.42528167366981506},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.32978662848472595},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.1887967586517334},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1621570587158203},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.16008102893829346}],"concepts":[{"id":"https://openalex.org/C2780130068","wikidata":"https://www.wikidata.org/wiki/Q2208651","display_name":"Transversal (combinatorics)","level":2,"score":0.95920729637146},{"id":"https://openalex.org/C158887663","wikidata":"https://www.wikidata.org/wiki/Q679367","display_name":"Latin square","level":4,"score":0.8829907178878784},{"id":"https://openalex.org/C2780428219","wikidata":"https://www.wikidata.org/wiki/Q16952335","display_name":"Cover (algebra)","level":2,"score":0.7628360986709595},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7354923486709595},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7242065668106079},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.6517934799194336},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.627597987651825},{"id":"https://openalex.org/C188721877","wikidata":"https://www.wikidata.org/wiki/Q103510","display_name":"Bar (unit)","level":2,"score":0.42528167366981506},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32978662848472595},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.1887967586517334},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1621570587158203},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.16008102893829346},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C153294291","wikidata":"https://www.wikidata.org/wiki/Q25261","display_name":"Meteorology","level":1,"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/C31903555","wikidata":"https://www.wikidata.org/wiki/Q1637030","display_name":"Food science","level":1,"score":0.0},{"id":"https://openalex.org/C70262065","wikidata":"https://www.wikidata.org/wiki/Q427652","display_name":"Rumen","level":3,"score":0.0},{"id":"https://openalex.org/C100544194","wikidata":"https://www.wikidata.org/wiki/Q41760","display_name":"Fermentation","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}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1007/s10623-018-0499-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10623-018-0499-9","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs10623-018-0499-9.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["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":"Designs, Codes and Cryptography","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1712.02460","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1712.02460","pdf_url":"https://arxiv.org/pdf/1712.02460","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:2775536822","is_oa":true,"landing_page_url":"https://arxiv.org/pdf/1712.02460.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:oai:monash.edu:openaire/7f06ba52-7a79-47e5-93de-1596dcde2526","is_oa":true,"landing_page_url":"https://research.monash.edu/en/publications/7f06ba52-7a79-47e5-93de-1596dcde2526","pdf_url":null,"source":{"id":"https://openalex.org/S4306402625","display_name":"Monash University Research Portal (Monash University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I56590836","host_organization_name":"Monash University","host_organization_lineage":["https://openalex.org/I56590836"],"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":"Best, D, Marbach, T, Stones, R J & Wanless, I M 2019, 'Covers and partial transversals of Latin squares', Designs Codes and Cryptography, vol. 87, no. 5, pp. 1109-1136. https://doi.org/10.1007/s10623-018-0499-9","raw_type":"article"},{"id":"pmh:oai:monash.edu:publications/7f06ba52-7a79-47e5-93de-1596dcde2526","is_oa":true,"landing_page_url":"http://www.scopus.com/inward/record.url?scp=85048764887&partnerID=8YFLogxK","pdf_url":null,"source":{"id":"https://openalex.org/S4306402625","display_name":"Monash University Research Portal (Monash University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I56590836","host_organization_name":"Monash University","host_organization_lineage":["https://openalex.org/I56590836"],"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":"Best , D , Marbach , T , Stones , R J &amp; Wanless , I M 2019 , ' Covers and partial transversals of Latin squares ' , Designs Codes and Cryptography , vol. 87 , no. 5 , pp. 1109-1136 . https://doi.org/10.1007/s10623-018-0499-9","raw_type":"article"},{"id":"doi:10.48550/arxiv.1712.02460","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1712.02460","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-journal"}],"best_oa_location":{"id":"doi:10.1007/s10623-018-0499-9","is_oa":true,"landing_page_url":"https://doi.org/10.1007/s10623-018-0499-9","pdf_url":"https://link.springer.com/content/pdf/10.1007%2Fs10623-018-0499-9.pdf","source":{"id":"https://openalex.org/S81616010","display_name":"Designs Codes and Cryptography","issn_l":"0925-1022","issn":["0925-1022","1573-7586"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319965","host_organization_name":"Springer Nature","host_organization_lineage":["https://openalex.org/P4310319965"],"host_organization_lineage_names":["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":"Designs, Codes and Cryptography","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1845600299","display_name":null,"funder_award_id":"NSERC CGS","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G2010946105","display_name":null,"funder_award_id":"NSERC CGS-D","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G2626896578","display_name":null,"funder_award_id":"Grant Number:","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2802911279","display_name":null,"funder_award_id":"Young","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3996507645","display_name":null,"funder_award_id":"CGS-D","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G4952545190","display_name":null,"funder_award_id":"Fellowship","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"},{"id":"https://openalex.org/G5854786607","display_name":null,"funder_award_id":"Tianjin","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G7608752429","display_name":null,"funder_award_id":"Talent","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2775536822.pdf","grobid_xml":"https://content.openalex.org/works/W2775536822.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W1493071618","https://openalex.org/W1606480398","https://openalex.org/W1852239058","https://openalex.org/W1966511824","https://openalex.org/W1985301313","https://openalex.org/W1988115705","https://openalex.org/W1997094995","https://openalex.org/W2010178857","https://openalex.org/W2017926358","https://openalex.org/W2037688303","https://openalex.org/W2042703645","https://openalex.org/W2045112111","https://openalex.org/W2075475446","https://openalex.org/W2313586032","https://openalex.org/W2320743697","https://openalex.org/W2322205785","https://openalex.org/W2557059219","https://openalex.org/W2601649413","https://openalex.org/W2964008621","https://openalex.org/W4238841606","https://openalex.org/W4254230493"],"related_works":["https://openalex.org/W3120529259","https://openalex.org/W2557059219","https://openalex.org/W2950778433","https://openalex.org/W2514574324","https://openalex.org/W1545563111","https://openalex.org/W1594805839","https://openalex.org/W2078241609","https://openalex.org/W2963507266","https://openalex.org/W2089750896","https://openalex.org/W2736654160","https://openalex.org/W2762677943","https://openalex.org/W3159230994","https://openalex.org/W70867195","https://openalex.org/W2000291348","https://openalex.org/W3084403602","https://openalex.org/W980436982","https://openalex.org/W2022253669","https://openalex.org/W2154892217","https://openalex.org/W2122595745","https://openalex.org/W1662501275"],"abstract_inverted_index":{"We":[0,73,82],"define":[1],"a":[2,5,10,42,91,94,110,124,148,152,172,176,197,204],"cover":[3,27,92,122,154,174],"of":[4,12,20,44,52,90,97,109,127,155,168,175,200,208,228],"Latin":[6,95,125,149,166,198,221],"square":[7,96,126,150,167,199],"to":[8,160,182],"be":[9],"set":[11,43],"entries":[13,45],"that":[14,46],"includes":[15,47],"at":[16,48,132],"least":[17],"one":[18,50],"representative":[19,51],"each":[21,53],"row,":[22,54],"column":[23,55],"and":[24,56,79,103,191],"symbol.":[25,57],"A":[26,38,58,120],"is":[28,41,61,65,100,113,179,196],"minimal":[29,121,153,173],"if":[30,63,102,105],"it":[31,64],"does":[32],"not":[33,66],"contain":[34],"any":[35,69,214],"smaller":[36],"cover.":[37],"partial":[39,59,71,80,111,206,226],"transversal":[40,60,112,207,227],"most":[49,133],"maximal":[62,205,225],"contained":[67],"in":[68,93,123],"larger":[70],"transversal.":[72],"explore":[74],"the":[75,84,106],"relationship":[76],"between":[77],"covers":[78],"transversals.":[81],"prove":[83],"following:":[85],"(1)":[86],"The":[87],"minimum":[88],"size":[89,108,131,157,177,209,229],"order":[98,128,169,201],"n":[99,129,143,159,170,202],"$$n+a$$":[101],"only":[104],"maximum":[107],"either":[114],"$$n-2a$$":[115],"or":[116],"$$n-2a+1$$":[117],".":[118,136,163,185,211,234],"(2)":[119],"has":[130,171],"$$\\mu":[134,161,183],"_n=3(n+1/2-\\sqrt{n+1/4})$$":[135],"(3)":[137],"There":[138],"are":[139],"infinitely":[140],"many":[141],"orders":[142],"for":[144],"which":[145,178],"there":[146,195],"exists":[147],"having":[151],"every":[156],"from":[158],"_n$$":[162,184],"(4)":[164],"Every":[165],"asymptotically":[180,218],"equal":[181],"(5)":[186],"If":[187],"$$1\\leqslant":[188],"k\\leqslant":[189],"n/2$$":[190],"$$n\\geqslant":[192],"5$$":[193],"then":[194],"with":[203],"$$n-k$$":[210],"(6)":[212],"For":[213],"$$\\varepsilon":[215],">0$$":[216],",":[217],"almost":[219],"all":[220],"squares":[222],"have":[223],"no":[224],"less":[230],"than":[231],"$$n-n^{2/3+\\varepsilon":[232],"}$$":[233]},"counts_by_year":[],"updated_date":"2026-03-18T14:38:29.013473","created_date":"2025-10-10T00:00:00"}
