{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,11,3]],"date-time":"2023-11-03T14:07:27Z","timestamp":1699020447198},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":15625,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1971,6]]},"abstract":"A significant portion of the study of large cardinals in set theory centers around the concept of \u201cpartition relation\u201d. To best capture the basic idea here, we introduce the following notation: for x<\/jats:italic> and y<\/jats:italic> sets, \u03ba<\/jats:italic> an infinite cardinal, and \u03b3<\/jats:italic> an ordinal less than \u03ba<\/jats:italic>, we let [x<\/jats:italic>]\u03b3<\/jats:italic><\/jats:sup> denote the collection of subsets of x<\/jats:italic> of order-type \u03b3<\/jats:italic> and abbreviate with the partition relation for each function F from<\/jats:italic>into y there exists a subset C of \u03ba of cardinality \u03ba such that<\/jats:italic> (such that for each \u03b1<\/jats:italic> < \u03b3<\/jats:italic>) the range of F on<\/jats:italic> [\u0421<\/jats:italic>]\u03b3<\/jats:italic><\/jats:sup> ([\u0421<\/jats:italic>]\u03b1<\/jats:italic><\/jats:sup>) has cardinality<\/jats:italic> 1. Now although each infinite cardinal \u03ba<\/jats:italic> satisfies the relation for each n<\/jats:italic> and m<\/jats:italic> in \u03c9<\/jats:italic> (F. P. Ramsey [8]), a connection with large cardinals arises when one asks, \u201cFor which uncountable \u03ba<\/jats:italic> do we have \u03ba<\/jats:italic> \u2192 (\u03ba<\/jats:italic>)2<\/jats:sup>?\u201d Indeed, any uncountable cardinal \u03ba<\/jats:italic> which satisfies \u03ba<\/jats:italic> \u2192 (\u03ba<\/jats:italic>)2<\/jats:sup> is strongly inaccessible and weakly compact (see [9]). As another example one can look at the improvements of Scott's original result to the effect that if there exists a measurable cardinal then there exists a nonconstructible set. Indeed, if \u03ba<\/jats:italic> is a measurable cardinal then \u03ba<\/jats:italic> \u2192 (\u03ba<\/jats:italic>)< \u03c9<\/jats:italic><\/jats:sup>, and as Solovay [11] has shown, if there exists a cardinal \u03ba<\/jats:italic> such that \u03ba<\/jats:italic> \u2192 (\u03ba<\/jats:italic>)< \u03c9<\/jats:italic>3<\/jats:sup> (\u03ba<\/jats:italic> \u2192 (\u2135<\/jats:bold>1<\/jats:sub>)< \u03c9<\/jats:italic><\/jats:sup>, even) then there exists a nonconstructible set of integers.<\/jats:p>","DOI":"10.2307\/2270265","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:07:49Z","timestamp":1146949669000},"page":"305-308","source":"Crossref","is-referenced-by-count":5,"title":["On large cardinals and partition relations"],"prefix":"10.1017","volume":"36","author":[{"given":"E. M.","family":"Kleinberg","sequence":"first","affiliation":[]},{"given":"R. A.","family":"Shore","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200083286_ref011","first-page":"50","article-title":"A nonconstructable set of integers","volume":"127","author":"Solovay","year":"1967","journal-title":"Transactions of the American Mathematical Society"},{"key":"S0022481200083286_ref010","first-page":"60","volume":"35","author":"Silver","year":"1970","journal-title":"Every analytic set is Ramsey"},{"key":"S0022481200083286_ref008","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s2-30.1.264"},{"key":"S0022481200083286_ref007","first-page":"931","article-title":"On a generalization of Ramsey's theorem","volume":"15","author":"Mathias","year":"1968","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200083286_ref006","first-page":"840","article-title":"Somewhat homogeneous sets II","volume":"16","author":"Kleinberg","year":"1969","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200083286_ref005","unstructured":"Kleinberg E. M. , Strong partition properties for infinite cardinals, this Journal (to appear)."},{"key":"S0022481200083286_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1956-10036-0"},{"key":"S0022481200083286_ref001","doi-asserted-by":"publisher","DOI":"10.1007\/BF02023868"},{"key":"S0022481200083286_ref009","volume-title":"Some applications of model theory in set theory","author":"Silver","year":"1966"},{"key":"S0022481200083286_ref004","unstructured":"Galvin F. and Prikry K. , to appear."},{"key":"S0022481200083286_ref003","first-page":"253","article-title":"A generalization of Ramsey's theorem","volume":"14","author":"Galvin","year":"1967","journal-title":"Notices of the American Mathematical Society"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200083286","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,31]],"date-time":"2019-05-31T20:05:36Z","timestamp":1559333136000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200083286\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1971,6]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1971,6]]}},"alternative-id":["S0022481200083286"],"URL":"https:\/\/doi.org\/10.2307\/2270265","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1971,6]]}}}