{"id":"https://openalex.org/W4401756370","doi":"https://doi.org/10.3390/sym16081092","title":"Extending Ramsey Numbers for Connected Graphs of Size 3","display_name":"Extending Ramsey Numbers for Connected Graphs of Size 3","publication_year":2024,"publication_date":"2024-08-22","ids":{"openalex":"https://openalex.org/W4401756370","doi":"https://doi.org/10.3390/sym16081092"},"language":"en","primary_location":{"id":"doi:10.3390/sym16081092","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym16081092","pdf_url":"https://www.mdpi.com/2073-8994/16/8/1092/pdf?version=1724325406","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/16/8/1092/pdf?version=1724325406","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5098710117","display_name":"Emma Jent","orcid":null},"institutions":[{"id":"https://openalex.org/I141649380","display_name":"Western Michigan University","ror":"https://ror.org/04j198w64","country_code":"US","type":"education","lineage":["https://openalex.org/I141649380"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Emma Jent","raw_affiliation_strings":["Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008-5248, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008-5248, USA","institution_ids":["https://openalex.org/I141649380"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5098727319","display_name":"Sawyer Osborn","orcid":null},"institutions":[{"id":"https://openalex.org/I141649380","display_name":"Western Michigan University","ror":"https://ror.org/04j198w64","country_code":"US","type":"education","lineage":["https://openalex.org/I141649380"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Sawyer Osborn","raw_affiliation_strings":["Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008-5248, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008-5248, USA","institution_ids":["https://openalex.org/I141649380"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5031137001","display_name":"Ping Zhang","orcid":null},"institutions":[{"id":"https://openalex.org/I141649380","display_name":"Western Michigan University","ror":"https://ror.org/04j198w64","country_code":"US","type":"education","lineage":["https://openalex.org/I141649380"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Ping Zhang","raw_affiliation_strings":["Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008-5248, USA"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008-5248, USA","institution_ids":["https://openalex.org/I141649380"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5031137001"],"corresponding_institution_ids":["https://openalex.org/I141649380"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.3713,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.76797274,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":91,"max":95},"biblio":{"volume":"16","issue":"8","first_page":"1092","last_page":"1092"},"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":0.9998000264167786,"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":0.9998000264167786,"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":0.9993000030517578,"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/T12541","display_name":"Graph Labeling and Dimension Problems","score":0.9987999796867371,"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/combinatorics","display_name":"Combinatorics","score":0.5225143432617188},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.49234139919281006},{"id":"https://openalex.org/keywords/ramseys-theorem","display_name":"Ramsey's theorem","score":0.48714134097099304},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3891531527042389},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.34971749782562256},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.20158088207244873}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5225143432617188},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.49234139919281006},{"id":"https://openalex.org/C44115641","wikidata":"https://www.wikidata.org/wiki/Q918099","display_name":"Ramsey's theorem","level":3,"score":0.48714134097099304},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3891531527042389},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34971749782562256},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.20158088207244873}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.3390/sym16081092","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym16081092","pdf_url":"https://www.mdpi.com/2073-8994/16/8/1092/pdf?version=1724325406","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:cf03ce0b52e54602810b6f536d223071","is_oa":true,"landing_page_url":"https://doaj.org/article/cf03ce0b52e54602810b6f536d223071","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 16, Iss 8, p 1092 (2024)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2073-8994/16/8/1092/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/sym16081092","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"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":"Symmetry","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/sym16081092","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym16081092","pdf_url":"https://www.mdpi.com/2073-8994/16/8/1092/pdf?version=1724325406","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4401756370.pdf","grobid_xml":"https://content.openalex.org/works/W4401756370.grobid-xml"},"referenced_works_count":13,"referenced_works":["https://openalex.org/W1487872891","https://openalex.org/W2007801688","https://openalex.org/W2045472305","https://openalex.org/W2066486518","https://openalex.org/W2095689360","https://openalex.org/W2318194046","https://openalex.org/W3135732667","https://openalex.org/W3154899610","https://openalex.org/W3204532083","https://openalex.org/W4239800961","https://openalex.org/W4367176385","https://openalex.org/W4392638328","https://openalex.org/W4400294108"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W1561903201","https://openalex.org/W2570467228","https://openalex.org/W2043020651","https://openalex.org/W2990545556","https://openalex.org/W2018966352","https://openalex.org/W1970497402","https://openalex.org/W2101126588","https://openalex.org/W2949154825","https://openalex.org/W4302078066"],"abstract_inverted_index":{"It":[0,38],"is":[1,36,39,85,125],"well":[2],"known":[3,41],"that":[4,42,92,204],"the":[5,12,23,26,80,86,97,100,120,126,137,148,169,175,187,192,196,219,231,236],"famous":[6],"Ramsey":[7,69,82,122,139,145,149],"number":[8,83,123,128],"R(K3,K3)=6.":[9],"That":[10],"is,":[11],"minimum":[13,87],"positive":[14,77,88,208],"integer":[15,78,89,209,215],"n":[16,90,106],"for":[17,129,156,168,186,206,213,245],"which":[18,55],"every":[19,43,93,207,214],"red-blue":[20,44,94],"coloring":[21,45,95],"of":[22,25,46,68,96,99,104,114,141,178,199,233],"edges":[24,98],"complete":[27,101],"graph":[28,73,102,161,247],"Kn":[29,103],"results":[30,48,107],"in":[31,49,108,160],"a":[32,72,76,153],"monochromatic":[33,53,112],"triangle":[34],"K3":[35,173],"6.":[37],"also":[40],"K6":[47],"at":[50],"least":[51],"two":[52,170],"triangles,":[54],"need":[56],"not":[57],"be":[58],"vertex-disjoint":[59,81,111],"or":[60],"edge-disjoint.":[61],"This":[62],"fact":[63],"led":[64],"to":[65,117],"an":[66],"extension":[67],"numbers.":[70],"For":[71,217],"F":[74],"and":[75,134,151,174,195,211,222,239],"t,":[79],"VRt(F)":[84,238],"such":[91],"order":[105,179],"t":[109,210,243],"pairwise":[110],"copies":[113],"subgraphs":[115],"isomorphic":[116],"F,":[118,142],"while":[119],"edge-disjoint":[121,130],"ERt(F)":[124,240],"corresponding":[127],"subgraphs.":[131],"Since":[132],"VR1(F)":[133],"ER1(F)":[135],"are":[136,224],"well-known":[138],"numbers":[140,150,164,185,220,227,237],"these":[143,184],"new":[144,154],"concepts":[146],"generalize":[147],"provide":[152,228],"perspective":[155],"this":[157],"classical":[158],"topic":[159],"theory.":[162],"These":[163,226],"have":[165],"been":[166],"investigated":[167],"connected":[171,189],"graphs":[172],"path":[176,193],"P3":[177],"3.":[180,201],"Here,":[181],"we":[182],"study":[183],"remaining":[188],"graphs,":[190],"namely,":[191],"P4":[194],"star":[197],"K1,3":[198],"size":[200],"We":[202],"show":[203],"VRt(P4)=4t+1":[205],"VRt(K1,3)=4t":[212],"t\u22652.":[216],"t\u22644,":[218],"ERt(K1,3)":[221],"ERt(P4)":[223],"determined.":[225],"information":[229],"towards":[230],"goal":[232],"determining":[234],"how":[235],"increase":[241],"as":[242],"increases":[244],"each":[246],"F\u2208{K1,3,P4}.":[248]},"counts_by_year":[{"year":2025,"cited_by_count":1}],"updated_date":"2026-03-11T06:11:40.159057","created_date":"2025-10-10T00:00:00"}