{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,20]],"date-time":"2023-10-20T05:36:42Z","timestamp":1697780202993},"reference-count":7,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2006,10,3]],"date-time":"2006-10-03T00:00:00Z","timestamp":1159833600000},"content-version":"vor","delay-in-days":8617,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1983,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Chv\u00e1tal established that <jats:italic>r<\/jats:italic>(<jats:italic>T<jats:sub>m<\/jats:sub>, K<jats:sub>n<\/jats:sub><\/jats:italic>) = (<jats:italic>m<\/jats:italic> \u2013 1)(<jats:italic>n<\/jats:italic> \u2013 1) + 1, where <jats:italic>T<jats:sub>m<\/jats:sub><\/jats:italic> is an arbitrary tree of order <jats:italic>m<\/jats:italic> and <jats:italic>K<jats:sub>n<\/jats:sub><\/jats:italic> is the complete graph of order <jats:italic>n<\/jats:italic>. This result was extended by Chartrand, Gould, and Polimeni who showed <jats:italic>K<jats:sub>n<\/jats:sub><\/jats:italic> could be replaced by a graph with clique number <jats:italic>n<\/jats:italic> and order <jats:italic>n<\/jats:italic> + 1 provided <jats:italic>n<\/jats:italic> \u2267 3 and <jats:italic>m<\/jats:italic> \u2267 3. We further extend these results to show that <jats:italic>K<jats:sub>n<\/jats:sub><\/jats:italic> can be replaced by any graph on <jats:italic>n<\/jats:italic> + 2 vertices with clique number <jats:italic>n<\/jats:italic>, provided <jats:italic>n<\/jats:italic> \u2267 5 and <jats:italic>m<\/jats:italic> \u2267 4. We then show that further extensions, in particular to graphs on <jats:italic>n<\/jats:italic> + 3 vertices with clique number <jats:italic>n<\/jats:italic> are impossible. We also investigate the Ramsey number of trees versus complete graphs minus sets of independent edges. We show that <jats:italic>r<\/jats:italic>(<jats:italic>T<jats:sub>m<\/jats:sub>, K<jats:sub>n<\/jats:sub> \u2013tK<\/jats:italic><jats:sub>2<\/jats:sub>) = (<jats:italic>m<\/jats:italic> \u2013 1)(<jats:italic>n \u2013 t<\/jats:italic> \u2013 1) + 1 for <jats:italic>m<\/jats:italic> \u2267 3, <jats:italic>n<\/jats:italic> \u2267 6, where <jats:italic>T<jats:sub>m<\/jats:sub><\/jats:italic> is any tree of order <jats:italic>m<\/jats:italic> except the star, and for each <jats:italic>t<\/jats:italic>, O \u2266 <jats:italic>t<\/jats:italic> \u2266 [(<jats:italic>n<\/jats:italic> \u2013 2)\/2].<\/jats:p>","DOI":"10.1002\/jgt.3190070109","type":"journal-article","created":{"date-parts":[[2007,5,29]],"date-time":"2007-05-29T06:56:37Z","timestamp":1180421797000},"page":"71-78","source":"Crossref","is-referenced-by-count":2,"title":["On the Ramsey number of trees versus graphs with large clique number"],"prefix":"10.1002","volume":"7","author":[{"given":"Ronald J.","family":"Gould","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michael S.","family":"Jacobson","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2006,10,3]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190010118"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.2140\/pjm.1972.41.335"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190040211"},{"key":"e_1_2_1_5_2","doi-asserted-by":"publisher","DOI":"10.1016\/0012-365X(74)90122-8"},{"key":"e_1_2_1_6_2","unstructured":"R. J.Faudree C. C.Rousseau andR. H.Schelp Generalizations of a ramsey result of Chv\u00e1tal.Proceedings of the Fourth International Conference on the Theory and Applications of Graphs Kalamazoo May1980 351\u2013361."},{"key":"e_1_2_1_7_2","doi-asserted-by":"publisher","DOI":"10.1002\/jgt.3190060405"},{"key":"e_1_2_1_8_2","unstructured":"M. S.Jacobson On Various Extensions of Ramsey Theory. Doctoral Dissertation Emory University 1980."}],"container-title":["Journal of Graph Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1002%2Fjgt.3190070109","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/pdf\/10.1002\/jgt.3190070109","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,19]],"date-time":"2023-10-19T22:46:43Z","timestamp":1697755603000},"score":1,"resource":{"primary":{"URL":"https:\/\/onlinelibrary.wiley.com\/doi\/10.1002\/jgt.3190070109"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1983,3]]},"references-count":7,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1983,3]]}},"alternative-id":["10.1002\/jgt.3190070109"],"URL":"https:\/\/doi.org\/10.1002\/jgt.3190070109","archive":["Portico"],"relation":{},"ISSN":["0364-9024","1097-0118"],"issn-type":[{"value":"0364-9024","type":"print"},{"value":"1097-0118","type":"electronic"}],"subject":[],"published":{"date-parts":[[1983,3]]}}}