{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T22:18:10Z","timestamp":1649197090389},"reference-count":4,"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":10876,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1984,6]]},"abstract":"<jats:p>Definition. For ordinals <jats:italic>\u03b1<\/jats:italic> \u2264 <jats:italic>\u03ba<\/jats:italic>, = [X]<jats:sup>\u03b1<\/jats:sup> = {p \u2286 X \u2223 ot (p) = <jats:italic>\u03b1<\/jats:italic>}. For <jats:italic>\u03b1<\/jats:italic> \u2264, <jats:italic>\u03ba, \u03ba<\/jats:italic> a cardinal, <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200033612_inline1\" \/> holds iff for every partition <jats:italic>F: [\u039a]<jats:sup>\u03ba<\/jats:sup> \u2192 A<\/jats:italic>, there is an <jats:italic>X<\/jats:italic> \u2208 [\u039a]<jats:sup>\u03ba<\/jats:sup> with <jats:italic>F<\/jats:italic> constant on [X]<jats:sup><jats:italic>\u03b1<\/jats:italic><\/jats:sup>. <jats:italic>X<\/jats:italic> is called <jats:italic>homogeneous<\/jats:italic> for F. When <jats:italic>A<\/jats:italic> = 2 the subscript is omitted.<\/jats:p><jats:p>It has been known since the sixties that for finite exponents all such properties are equivalent, i.e., <jats:italic>\u03ba<\/jats:italic>\u2192 (<jats:italic>\u03ba<\/jats:italic>)<jats:sup>2<\/jats:sup> iff <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200033612_inline2\" \/> for all <jats:italic>n<\/jats:italic> &lt; <jats:italic>\u03c9, \u03b2<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic>. The infinite case is much more difficult, though it is not actually known to be different. The usual methods for dealing with partitions fail, and as the weakest of these properties violates the Axiom of Choice, many techniques are unavailable.<\/jats:p><jats:p>It is extremely unlikely that an infinite exponent can be increased, but infinite subscripts are generally possible.<\/jats:p><jats:p>Theorem. <jats:italic>If<\/jats:italic> \u2225 <jats:italic>\u03b1<\/jats:italic> \u2225 \u00b7 2 \u2264 <jats:italic>\u03b1<\/jats:italic>, then \u03ba \u2192 (<jats:italic>\u03ba<\/jats:italic>)<jats:sup><jats:italic>\u03b1<\/jats:italic><\/jats:sup> implies <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200033612_inline3\" \/> for all <jats:italic>\u03bb<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic>.<\/jats:p><jats:p>In \u00a71, we show the subscript can be any <jats:italic>\u03bb<\/jats:italic> &lt; <jats:italic>\u03ba<\/jats:italic>. In \u00a72 we extend this to 2<jats:sup><jats:italic>\u03bb<\/jats:italic><\/jats:sup>. In \u00a73 we discuss the possibilities with limited amounts of well-ordered choice, and apply the theorem to the problem of obliging ordinals. Except in \u00a73, we assume no choice. For the story of finite exponents, see [1]. For the relationship between infinite-exponent partition relations and choice, see [4].<\/jats:p>","DOI":"10.2307\/2274188","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:08:28Z","timestamp":1146953308000},"page":"558-562","source":"Crossref","is-referenced-by-count":1,"title":["Infinite subscripts from infinite exponents"],"prefix":"10.1017","volume":"49","author":[{"given":"James E.","family":"Baumgartner","sequence":"first","affiliation":[]},{"given":"James M.","family":"Henle","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200033612_ref003","first-page":"410","volume":"35","author":"Kleinberg","year":"1970","journal-title":"Strong partition properties for infinite cardinals"},{"key":"S0022481200033612_ref001","volume-title":"Set theory","author":"Drake","year":"1974"},{"key":"S0022481200033612_ref004","first-page":"299","volume":"38","author":"Kleinberg","year":"1973","journal-title":"Infinite-exponent partition relations and well-ordered choice"},{"key":"S0022481200033612_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760831"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200033612","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T20:48:42Z","timestamp":1558644522000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200033612\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1984,6]]},"references-count":4,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1984,6]]}},"alternative-id":["S0022481200033612"],"URL":"https:\/\/doi.org\/10.2307\/2274188","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1984,6]]}}}