{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,4]],"date-time":"2022-04-04T21:34:52Z","timestamp":1649108092212},"reference-count":9,"publisher":"Cambridge University Press (CUP)","issue":"4","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":10693,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1984,12]]},"abstract":"<jats:p>A strong partition cardinal is an uncountable well-ordered cardinal <jats:italic>\u03ba<\/jats:italic> such that every partition of [<jats:italic>\u03ba<\/jats:italic>]<jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup> (the size <jats:italic>\u03ba<\/jats:italic> subsets of <jats:italic>\u03ba<\/jats:italic>) into less than <jats:italic>\u03ba<\/jats:italic> many pieces has a homogeneous set of size \u03ba. The existence of such cardinals is inconsistent with the axiom of choice, and our work concerning them is carried out in ZF set theory with just dependent choice (DC). The consistency of strong partition cardinals with this weaker theory remains an open question. The axiom of determinacy (AD) implies that a large number of cardinals including \u2135<jats:sub>1<\/jats:sub> have the strong partition property. The hypothesis that AD holds in the inner model of constructible sets built over the real numbers as urelements has important consequences for descriptive set theory, and results concerning strong partition cardinals are often applied in this context. Kechris [4] and Kechris et al. [5] contain further information concerning the relationship between AD and strong partition cardinals.<\/jats:p><jats:p>We assume familiarity with the basic results on strong partition cardinals as developed in Kleinberg [6], [7], [8] and Henle [2]. Recall that a strong partition cardinal <jats:italic>\u03ba<\/jats:italic> is measurable; in fact every stationary subset of <jats:italic>\u03ba<\/jats:italic> is measure one under some normal measure on <jats:italic>\u03ba<\/jats:italic>. If <jats:italic>\u03bc<\/jats:italic> is a countably additive ultrafilter extending the closed unbounded filter on <jats:italic>\u03ba<\/jats:italic>, then the length of the ultrapower [<jats:italic>\u03ba<\/jats:italic>]<jats:sup><jats:italic>\u03ba<\/jats:italic><\/jats:sup> under the less than almost everywhere <jats:italic>\u03bc<\/jats:italic> ordering is again a measurable cardinal. In \u00a71 below we establish a polarized partition property on these measurable cardinals.<\/jats:p>","DOI":"10.2307\/2274277","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:10:48Z","timestamp":1146953448000},"page":"1268-1272","source":"Crossref","is-referenced-by-count":0,"title":["On the ultrafilters and ultrapowers of strong partition cardinals"],"prefix":"10.1017","volume":"49","author":[{"given":"J.M.","family":"Henle","sequence":"first","affiliation":[]},{"given":"E.M.","family":"Kleinberg","sequence":"additional","affiliation":[]},{"given":"R.J.","family":"Watro","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200042523_ref008","first-page":"161","volume":"47","author":"Kleinberg","year":"1982","journal-title":"A measure representation theorem for strong partition cardinals"},{"key":"S0022481200042523_ref007","first-page":"643","volume":"46","author":"Kleinberg","year":"1981","journal-title":"Producing measurable cardinals beyond k"},{"key":"S0022481200042523_ref006","volume-title":"Infinitary combinatorics and the axiom of determinateness","volume":"612","author":"Kleinberg","year":"1979"},{"key":"S0022481200042523_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0090236"},{"key":"S0022481200042523_ref004","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0069296"},{"key":"S0022481200042523_ref003","volume-title":"Proceedings of the Sixth Latin American Symposium on Mathematical Logic (Caracas, 1983)","author":"Henle"},{"key":"S0022481200042523_ref001","unstructured":"Becker H. , Review of [6], this Journal, vol. 47 (1982), pp. 698\u2013701."},{"key":"S0022481200042523_ref009","unstructured":"Watro R. J. , Effects of infinite exponent partition properties on Mahlo cardinals, Ph.D. thesis, State University of New York at Buffalo, Buffalo, New York, 1982."},{"key":"S0022481200042523_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(79)90024-X"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200042523","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T19:11:20Z","timestamp":1558638680000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200042523\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1984,12]]},"references-count":9,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1984,12]]}},"alternative-id":["S0022481200042523"],"URL":"https:\/\/doi.org\/10.2307\/2274277","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1984,12]]}}}