{"id":"https://openalex.org/W41590842","doi":"https://doi.org/10.37236/194","title":"Generalized Schur Numbers for $x_1 + x_2 + c = 3x_3$","display_name":"Generalized Schur Numbers for $x_1 + x_2 + c = 3x_3$","publication_year":2009,"publication_date":"2009-08-14","ids":{"openalex":"https://openalex.org/W41590842","doi":"https://doi.org/10.37236/194","mag":"41590842"},"language":"en","primary_location":{"id":"doi:10.37236/194","is_oa":true,"landing_page_url":"https://doi.org/10.37236/194","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r105/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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r105/pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5039000136","display_name":"Andr\u00e9 E. K\u00e9zdy","orcid":"https://orcid.org/0000-0002-5389-6758"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Andr\u00e9 E. K\u00e9zdy","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5040831057","display_name":"Hunter S. Snevily","orcid":null},"institutions":[{"id":"https://openalex.org/I155093810","display_name":"University of Idaho","ror":"https://ror.org/03hbp5t65","country_code":"US","type":"education","lineage":["https://openalex.org/I155093810"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Hunter S. Snevily","raw_affiliation_strings":["Department of Mathematics University of Idaho Moscow, ID 83844 USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics University of Idaho Moscow, ID 83844 USA","institution_ids":["https://openalex.org/I155093810"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5083874908","display_name":"Susan C. White","orcid":"https://orcid.org/0009-0009-7339-7997"},"institutions":[{"id":"https://openalex.org/I142740786","display_name":"University of Louisville","ror":"https://ror.org/01ckdn478","country_code":"US","type":"education","lineage":["https://openalex.org/I142740786"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Susan C. White","raw_affiliation_strings":["Department of Mathematics University of Louisville Louisville, KY 40292 USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics University of Louisville Louisville, KY 40292 USA","institution_ids":["https://openalex.org/I142740786"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5039000136"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.00418316,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"16","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9994999766349792,"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"}},"topics":[{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9994999766349792,"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"}},{"id":"https://openalex.org/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.9976999759674072,"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/T11428","display_name":"Advanced Mathematical Identities","score":0.9969000220298767,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/mathematics","display_name":"Mathematics","score":0.8477991819381714},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7763122320175171},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.32108235359191895}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8477991819381714},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7763122320175171},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32108235359191895}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/194","is_oa":true,"landing_page_url":"https://doi.org/10.37236/194","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r105/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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.214.4374","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.214.4374","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/Volume_16/PDF/v16i1r105.pdf","raw_type":"text"},{"id":"mag:41590842","is_oa":true,"landing_page_url":"https://dblp.uni-trier.de/db/journals/combinatorics/combinatorics16.html#KezdySW09","pdf_url":null,"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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":null}],"best_oa_location":{"id":"doi:10.37236/194","is_oa":true,"landing_page_url":"https://doi.org/10.37236/194","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v16i1r105/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":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W41590842.pdf","grobid_xml":"https://content.openalex.org/works/W41590842.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W6605110756"],"related_works":["https://openalex.org/W16323757","https://openalex.org/W3030936517","https://openalex.org/W2041678571","https://openalex.org/W2408474202","https://openalex.org/W2604855981","https://openalex.org/W2322240724","https://openalex.org/W2964198025","https://openalex.org/W2066048608","https://openalex.org/W1024647437","https://openalex.org/W3184451424","https://openalex.org/W3150276120","https://openalex.org/W2734580446","https://openalex.org/W2073603352","https://openalex.org/W2008262009","https://openalex.org/W3082595352","https://openalex.org/W2567695433","https://openalex.org/W2900456739","https://openalex.org/W2950381960","https://openalex.org/W1970613794","https://openalex.org/W2140517298"],"abstract_inverted_index":{"Let":[0],"$r(c)$":[1],"be":[2],"the":[3,14],"least":[4],"positive":[5],"integer":[6],"$n$":[7],"such":[8],"that":[9,38],"every":[10],"two":[11],"coloring":[12],"of":[13,32],"integers":[15],"$1,\\ldots,n$":[16],"contains":[17],"a":[18,30],"monochromatic":[19],"solution":[20],"to":[21],"$x_1":[22],"+":[23,25,43],"x_2":[24],"c":[26],"=":[27,41,54],"3x_3$.":[28],"Verifying":[29],"conjecture":[31],"Martinelli":[33],"and":[34,51],"Schaal,":[35],"we":[36],"prove":[37],"$$":[39,45,52,58],"r(c)":[40,53],"\\left\\lceil{2\\lceil{2+c\\over3}\\rceil":[42],"c\\over3}\\right\\rceil,":[44],"for":[46,59],"all":[47,60],"$c":[48,61],"\\ge":[49],"13$,":[50],"\\left\\lceil{3\\lceil{3-c\\over2}\\rceil":[55],"-":[56],"c\\over2}\\right\\rceil,":[57],"\\le":[62],"-4$.":[63]},"counts_by_year":[],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}