{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T11:57:00Z","timestamp":1697111820004},"reference-count":6,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":6401,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1996,9]]},"abstract":"<jats:p>Let <jats:italic>\u03ba<\/jats:italic> be an uncountable cardinal and the edges of a complete graph with <jats:italic>\u03ba<\/jats:italic> vertices be colored with \u2135<jats:sub>0<\/jats:sub> colors. For <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017114_inline1\" \/> the Erd\u0151s-Rado theorem implies that there is an infinite monochromatic subgraph. However, if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017114_inline2\" \/>, then it may be impossible to find a monochromatic triangle. This paper is concerned with the latter situation. We consider the types of colorings of finite subgraphs that must occur when the edges of the complete graph on <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017114_inline2\" \/> vertices are colored with \u2135<jats:sub>0<\/jats:sub> colors. In particular, we are concerned with the case \u2135<jats:sub>1<\/jats:sub> \u2264 <jats:italic>\u03ba<\/jats:italic> \u2264 \u2135<jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub>.<\/jats:p><jats:p>The study of these color patterns (known as identities) has a history that involves the existence of compactness theorems for two cardinal models [2]. When the graph being colored has size \u2135<jats:sub>1<\/jats:sub>, the identities that must occur have been classified by Shelah [4]. If the graph has size greater than or equal to \u2135<jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub> the identities have also been classified in [3]. The number of colors is fixed at \u2135<jats:sub>0<\/jats:sub> as it is the natural place to start and the results here can be generalized to situations where more colors are used.<\/jats:p><jats:p>There is one difference that we now make explicit. When countably many colors are used we can define the following coloring of the complete graph on <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200017114_inline3\" \/> vertices. First consider the branches in the complete binary tree of height <jats:italic>\u03c9<\/jats:italic> to be vertices of a complete graph.<\/jats:p>","DOI":"10.2307\/2275784","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:59:03Z","timestamp":1146941943000},"page":"780-787","source":"Crossref","is-referenced-by-count":3,"title":["Identities on cardinals less than \u2135<sub><i>\u03c9<\/i><\/sub>"],"prefix":"10.1017","volume":"61","author":[{"given":"M.","family":"Gilchrist","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"S.","family":"Shelah","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200017114_ref002","first-page":"177","article-title":"Transfer theorems and their applications to logics","author":"Schmerl","year":"1985","journal-title":"Model theoretic logics"},{"key":"S0022481200017114_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90009-8"},{"key":"S0022481200017114_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(76)90018-8"},{"key":"S0022481200017114_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1977-0434800-6"},{"key":"S0022481200017114_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(78)90017-7"},{"key":"S0022481200017114_ref006","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(87)90015-7"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200017114","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,12]],"date-time":"2019-05-12T16:33:31Z","timestamp":1557678811000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200017114\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1996,9]]},"references-count":6,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1996,9]]}},"alternative-id":["S0022481200017114"],"URL":"https:\/\/doi.org\/10.2307\/2275784","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1996,9]]}}}