{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,3]],"date-time":"2026-03-03T06:32:27Z","timestamp":1772519547828,"version":"3.50.1"},"reference-count":4,"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":8593,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,9]]},"abstract":"<jats:p>In [3], Todor\u010devi\u0107 showed that <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> \u21f8 [<jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>]<jats:sub arrange=\"stack\"><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub><jats:sup arrange=\"stack\">2<\/jats:sup>. In this paper we use similar methods to prove an analogous partition theorem for <jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic><\/jats:sub>1(<jats:italic>\u03bb<\/jats:italic>), for certain uncountable cardinals <jats:italic>\u03bb<\/jats:italic>.<\/jats:p><jats:p>Recall that <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> \u2192 [<jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>]<jats:sub arrange=\"stack\"><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub><jats:sup arrange=\"stack\">2<\/jats:sup>, means that for every function <jats:italic>f<\/jats:italic>: [<jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>]<jats:sup>2<\/jats:sup> \u2192 <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> there is a set <jats:italic>A<\/jats:italic> \u2208 [<jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>]<jats:sup><jats:italic>\u03c9<\/jats:italic><\/jats:sup><jats:sup>1<\/jats:sup> such that <jats:italic>f<\/jats:italic>\u201c[<jats:italic>A<\/jats:italic>]<jats:sup>2<\/jats:sup> \u2260 <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>, and of course <jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub> \u21f8 [<jats:italic>\u03c9<\/jats:italic><jats:sub>1<\/jats:sub>]<jats:sub arrange=\"stack\"><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub><jats:sup arrange=\"stack\">2<\/jats:sup>, is the negation of this statement. For partition relations on  <jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub>(<jats:italic>\u2192<\/jats:italic>) it is customary to partition only those pairs of sets in which the first set is a subset of the second. Thus for <jats:italic>A<\/jats:italic> \u2286 <jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub>(<jats:italic>\u03bb<\/jats:italic>) we define<\/jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200025524_eqnU1\"\/><\/jats:disp-formula><\/jats:p><jats:p>We will write <jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub>(<jats:italic>\u03bb<\/jats:italic>) \u2192 [unbdd]<jats:sub arrange=\"stack\"><jats:italic>\u03bb<\/jats:italic><\/jats:sub><jats:sup arrange=\"stack\">2<\/jats:sup> to mean that for every function <jats:italic>f<\/jats:italic>: [<jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub>(<jats:italic>\u03bb<\/jats:italic>)]<jats:sub arrange=\"stack\">\u2282<\/jats:sub><jats:sup arrange=\"stack\">2<\/jats:sup> \u2192 <jats:italic>\u03bb<\/jats:italic> there is an unbounded set <jats:italic>A<\/jats:italic> \u2286 <jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub>(<jats:italic>\u03bb<\/jats:italic>) such that <jats:italic>f<\/jats:italic>\u201c[<jats:italic>A<\/jats:italic>]<jats:sub arrange=\"stack\">\u2282<\/jats:sub><jats:sup arrange=\"stack\">2<\/jats:sup> \u2260 <jats:italic>\u03bb<\/jats:italic>, and again <jats:italic>P<\/jats:italic><jats:sub><jats:italic>\u03c9<\/jats:italic>1<\/jats:sub>(<jats:italic>\u03bb<\/jats:italic>) \u21f8 [unbdd]<jats:sub arrange=\"stack\"><jats:italic>\u03bb<\/jats:italic><\/jats:sub><jats:sup arrange=\"stack\">2<\/jats:sup> is the negation of this statement.<\/jats:p>","DOI":"10.2307\/2274470","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:36:50Z","timestamp":1146955010000},"page":"1019-1021","source":"Crossref","is-referenced-by-count":4,"title":["Partitioning pairs of countable sets of ordinals"],"prefix":"10.1017","volume":"55","author":[{"given":"Dan","family":"Velleman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200025524_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(73)90014-4"},{"key":"S0022481200025524_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(87)90018-2"},{"key":"S0022481200025524_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392561"},{"key":"S0022481200025524_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/031\/763904"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200025524","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T20:25:40Z","timestamp":1558211140000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200025524\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,9]]},"references-count":4,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1990,9]]}},"alternative-id":["S0022481200025524"],"URL":"https:\/\/doi.org\/10.2307\/2274470","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,9]]}}}