{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,3]],"date-time":"2025-11-03T09:01:34Z","timestamp":1762160494792},"reference-count":4,"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 subset <jats:italic>X<\/jats:italic> of a structure <jats:italic>S<\/jats:italic> is called <jats:italic>free<\/jats:italic> in <jats:italic>S<\/jats:italic> if \u2200<jats:italic>x<\/jats:italic> \u2208 <jats:italic>X<\/jats:italic><jats:italic>x<\/jats:italic> \u2209 <jats:italic>S<\/jats:italic>[<jats:italic>X<\/jats:italic> \u2212 {<jats:italic>x<\/jats:italic>}]; here, <jats:italic>S<\/jats:italic>[<jats:italic>Y<\/jats:italic>] is the substructure of <jats:italic>S<\/jats:italic> generated from <jats:italic>Y<\/jats:italic> by the functions of <jats:italic>S<\/jats:italic>. For <jats:italic>\u03ba, \u03bb, \u03bc<\/jats:italic> cardinals, let Fr<jats:sub><jats:italic>\u03bc<\/jats:italic><\/jats:sub>(<jats:italic>\u03ba, \u03bb<\/jats:italic>) be the assertion:<\/jats:p><jats:p><jats:disp-quote><jats:p>for every structure <jats:italic>S<\/jats:italic> with <jats:italic>\u03ba<\/jats:italic> \u2282 <jats:italic>S<\/jats:italic> which has at most <jats:italic>\u03bc<\/jats:italic> functions and relations there is a subset <jats:italic>X<\/jats:italic> \u2282 <jats:italic>\u03ba<\/jats:italic> free in <jats:italic>S<\/jats:italic> of cardinality \u2265 <jats:italic>\u03bb<\/jats:italic>.<\/jats:p><\/jats:disp-quote><\/jats:p><jats:p>We show that Fr<jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub>(<jats:italic>\u03c9<jats:sub>\u03c9<\/jats:sub>, \u03c9<\/jats:italic>), the <jats:italic>free-subset property<\/jats:italic> for <jats:italic>\u03c9<jats:sub>\u03c9<\/jats:sub><\/jats:italic>, is equiconsistent with the existence of a measurable cardinal (2.2,4.4). This answers a question of Devlin [De].<\/jats:p><jats:p>In the first section of this paper we prove some combinatorial facts about Fr<jats:sub><jats:italic>\u03bc<\/jats:italic><\/jats:sub>(<jats:italic>\u03ba, \u03bb<\/jats:italic>); in particular the first cardinal <jats:italic>\u03ba<\/jats:italic> such that Fr<jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub>(<jats:italic>\u03ba, \u03c9<\/jats:italic>) is weakly inaccessible or of cofinality <jats:italic>\u03c9<\/jats:italic> (1.2). The second section shows that, under Fr<jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub>(<jats:italic>\u03c9<jats:sub>\u03c9<\/jats:sub>, \u03c9<\/jats:italic>), <jats:italic>\u03c9<jats:sub>\u03c9<\/jats:sub><\/jats:italic> is measurable in an inner model. For the convenience of readers not acquainted with the core model <jats:italic>\u03ba<\/jats:italic>, we first deduce the existence of 0<jats:sup><jats:italic>#<\/jats:italic><\/jats:sup> (2.1) using the inner model <jats:italic>L<\/jats:italic>. Then we adapt the proof to the core model and obtain that <jats:italic>\u03c9<jats:sub>\u03c9<\/jats:sub><\/jats:italic> is measurable in an inner model. For the reverse direction, we essentially apply a construction of Shelah [Sh] who forced Fr<jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub>(<jats:italic>\u03c9<jats:sub>\u03c9<\/jats:sub>, \u03c9<\/jats:italic>) over a ground model which contains an \u03c9-sequence of measurable cardinals. We show in \u00a74 that indeed a <jats:italic>coherent sequence of Ramsey cardinals<\/jats:italic> suffices. In \u00a73 we obtain such a sequence as an endsegment of a Prikry sequence.<\/jats:p>","DOI":"10.2307\/2274272","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:10:48Z","timestamp":1146939048000},"page":"1198-1204","source":"Crossref","is-referenced-by-count":17,"title":["The consistency strength of the free-subset property for \u03c9<sub>\u03c9<\/sub>"],"prefix":"10.1017","volume":"49","author":[{"given":"Peter","family":"Koepke","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S002248120004247X_ref003","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511600586"},{"key":"S002248120004247X_ref004","first-page":"505","volume":"45","author":"Shelah","year":"1980","journal-title":"Independence of strong partition relation for small cardinals, and the free-subset problem"},{"key":"S002248120004247X_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(73)90010-7"},{"key":"S002248120004247X_ref002","first-page":"115","volume-title":"ISILC Logic Conference, Kiel 1974","volume":"499","author":"Devlin","year":"1975"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S002248120004247X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T15:10:07Z","timestamp":1558624207000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S002248120004247X\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1984,12]]},"references-count":4,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1984,12]]}},"alternative-id":["S002248120004247X"],"URL":"https:\/\/doi.org\/10.2307\/2274272","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1984,12]]}}}