{"id":"https://openalex.org/W1560624708","doi":"https://doi.org/10.37236/971","title":"Two Color Off-diagonal Rado-type Numbers","display_name":"Two Color Off-diagonal Rado-type Numbers","publication_year":2007,"publication_date":"2007-08-04","ids":{"openalex":"https://openalex.org/W1560624708","doi":"https://doi.org/10.37236/971","mag":"1560624708"},"language":"en","primary_location":{"id":"doi:10.37236/971","is_oa":true,"landing_page_url":"https://doi.org/10.37236/971","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v14i1r53/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/v14i1r53/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5056980575","display_name":"Kellen Myers","orcid":null},"institutions":[{"id":"https://openalex.org/I39660569","display_name":"Colgate University","ror":"https://ror.org/05d23ve83","country_code":"US","type":"education","lineage":["https://openalex.org/I39660569"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Kellen Myers","raw_affiliation_strings":["Department of Mathematics Colgate University, Hamilton, NY, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics Colgate University, Hamilton, NY, USA","institution_ids":["https://openalex.org/I39660569"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5009823885","display_name":"Aaron Robertson","orcid":null},"institutions":[{"id":"https://openalex.org/I39660569","display_name":"Colgate University","ror":"https://ror.org/05d23ve83","country_code":"US","type":"education","lineage":["https://openalex.org/I39660569"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Aaron Robertson","raw_affiliation_strings":["Department of Mathematics Colgate University, Hamilton, NY, USA","Colgate University,,"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics Colgate University, Hamilton, NY, USA","institution_ids":["https://openalex.org/I39660569"]},{"raw_affiliation_string":"Colgate University,,","institution_ids":["https://openalex.org/I39660569"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5056980575"],"corresponding_institution_ids":["https://openalex.org/I39660569"],"apc_list":null,"apc_paid":null,"fwci":0.4432,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.57943303,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"14","issue":"1","first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9986000061035156,"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"}},"topics":[{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9986000061035156,"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"}},{"id":"https://openalex.org/T11329","display_name":"Limits and Structures in Graph Theory","score":0.9966999888420105,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9919000267982483,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8400182723999023},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.8161120414733887},{"id":"https://openalex.org/keywords/diagonal","display_name":"Diagonal","score":0.785722017288208},{"id":"https://openalex.org/keywords/monochromatic-color","display_name":"Monochromatic color","score":0.7694526314735413},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.5888171195983887},{"id":"https://openalex.org/keywords/sign","display_name":"Sign (mathematics)","score":0.5174742341041565},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.5088046193122864},{"id":"https://openalex.org/keywords/homogeneous","display_name":"Homogeneous","score":0.47563791275024414},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.16713130474090576},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12517529726028442},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.12111929059028625}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8400182723999023},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.8161120414733887},{"id":"https://openalex.org/C130367717","wikidata":"https://www.wikidata.org/wiki/Q189791","display_name":"Diagonal","level":2,"score":0.785722017288208},{"id":"https://openalex.org/C40833965","wikidata":"https://www.wikidata.org/wiki/Q6901438","display_name":"Monochromatic color","level":2,"score":0.7694526314735413},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.5888171195983887},{"id":"https://openalex.org/C139676723","wikidata":"https://www.wikidata.org/wiki/Q1193832","display_name":"Sign (mathematics)","level":2,"score":0.5174742341041565},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5088046193122864},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.47563791275024414},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.16713130474090576},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12517529726028442},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.12111929059028625},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/971","is_oa":true,"landing_page_url":"https://doi.org/10.37236/971","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v14i1r53/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:10.1.1.178.5348","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.178.5348","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_14/PDF/v14i1r53.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.236.1205","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.236.1205","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/math/0606451v1.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/971","is_oa":true,"landing_page_url":"https://doi.org/10.37236/971","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v14i1r53/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":[{"id":"https://openalex.org/F4320310994","display_name":"Colgate University","ror":"https://ror.org/05d23ve83"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1560624708.pdf","grobid_xml":"https://content.openalex.org/works/W1560624708.grobid-xml"},"referenced_works_count":3,"referenced_works":["https://openalex.org/W1493267044","https://openalex.org/W1964974890","https://openalex.org/W2070496958"],"related_works":["https://openalex.org/W2509614667","https://openalex.org/W2963293152","https://openalex.org/W4302199713","https://openalex.org/W4282026465","https://openalex.org/W2626027010","https://openalex.org/W2963562790","https://openalex.org/W2909469610","https://openalex.org/W2755006503","https://openalex.org/W1629841509","https://openalex.org/W1547276641"],"abstract_inverted_index":{"We":[0,48,70,109],"show":[1],"that":[2,64,115],"for":[3,75,119,137],"any":[4,25],"two":[5,134],"linear":[6],"homogeneous":[7],"equations":[8],"${\\cal":[9,37,46,84],"E}_0,{\\cal":[10,56,77,100],"E}_1$,":[11],"each":[12,83,103],"with":[13],"at":[14],"least":[15],"three":[16],"variables":[17],"and":[18,128],"coefficients":[19],"not":[20],"all":[21],"the":[22,50,60,88,95,106],"same":[23],"sign,":[24],"2-coloring":[26],"of":[27,33,42,87,98,105,123],"${\\Bbb":[28],"Z}^+$":[29],"admits":[30],"monochromatic":[31,40],"solutions":[32,41],"color":[34,43],"0":[35],"to":[36,45,58,131],"E}_0$":[38],"or":[39],"1":[44],"E}_1$.":[47],"define":[49],"2-color":[51],"off-diagonal":[52,120],"Rado":[53,121,139],"number":[54],"$RR({\\cal":[55,76,99],"E}_1)$":[57,78,101],"be":[59],"smallest":[61],"$N$":[62],"such":[63,68],"$[1,N]$":[65],"must":[66],"admit":[67],"solutions.":[69],"determine":[71],"a":[72,112,124],"lower":[73],"bound":[74],"in":[79],"certain":[80],"cases":[81],"when":[82,102],"E}_i$":[85],"is":[86,104],"form":[89,107],"$a_1x_1+\\dots+a_nx_n=z$":[90],"as":[91,93],"well":[92],"find":[94],"exact":[96],"value":[97],"$x_1+a_2x_2+\\dots+a_nx_n=z$.":[108],"then":[110],"present":[111],"Maple":[113],"package":[114],"determines":[116],"upper":[117],"bounds":[118],"numbers":[122],"few":[125],"particular":[126],"types,":[127],"use":[129],"it":[130],"quickly":[132],"prove":[133],"previous":[135],"results":[136],"diagonal":[138],"numbers.":[140]},"counts_by_year":[{"year":2013,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}