{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,10]],"date-time":"2026-03-10T13:10:20Z","timestamp":1773148220331,"version":"3.50.1"},"reference-count":13,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":8777,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1990,3]]},"abstract":"<jats:p>For e.g. <jats:italic>\u03bb<\/jats:italic> = <jats:italic>\u03bc<\/jats:italic><jats:sup>+<\/jats:sup>, <jats:italic>\u03bc<\/jats:italic> regular, <jats:italic>\u03bb<\/jats:italic> larger than the continuum, we prove a strong nonpartition result (stronger than <jats:italic>\u03bb<\/jats:italic> \u2192 [<jats:italic>\u03bb; \u03bb<\/jats:italic>]<jats:sup>2<\/jats:sup>). As a consequence, the product of two topological spaces of cellularity &lt;<jats:italic>\u03bb<\/jats:italic> may have cellularity <jats:italic>\u03bb<\/jats:italic>, or, in equivalent formulation, the product of two <jats:italic>\u03bb<\/jats:italic>-c.c. Boolean algebras may lack the <jats:italic>\u03bb<\/jats:italic>-c.c. Also <jats:italic>\u03bb<\/jats:italic>-S-spaces and <jats:italic>\u03bb<\/jats:italic>-L-spaces exist. In fact we deal not with successors of regular <jats:italic>\u03bb<\/jats:italic> but with regular <jats:italic>\u03bb<\/jats:italic> above the continuum which has a nonreflecting stationary subset of ordinals with uncountable cofinalities; sometimes we require <jats:italic>\u03bb<\/jats:italic> to be not strong limit.<\/jats:p><jats:p>The paper is self-contained. On the nonpartition results see the closely related papers of Todor\u010devi\u0107 [T1], Shelah [Sh276] and [Sh261], and Shelah and Steprans [ShSt1].<\/jats:p><jats:p>On the cellularity of products see Todor\u010devi\u0107 [T2] and [T3], where such results were obtained for (e.g.) cf <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026402_inline1\"\/> and <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026402_inline2\"\/>; the class of cardinals he gets is quite disjoint from ours. In [Sh282] such results were obtained for more successors of singulars (mainly <jats:italic>\u03bb<\/jats:italic><jats:sup>+<\/jats:sup>, <jats:italic>\u03bb<\/jats:italic> &gt; 2<jats:sup>cf <jats:italic>\u03bb<\/jats:italic><\/jats:sup>). Also, concerning <jats:italic>S<\/jats:italic> and <jats:italic>L<\/jats:italic> spaces, Todor\u010devi\u0107 gets existence.<\/jats:p><jats:p>Todor\u010devi\u0107's work on cardinals like <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026402_inline2\"\/> relies on [Sh68] (see more in [ShA2, Chapter XIII]) (the scales appearing in the proof of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200026402_inline3\"\/>). The problem was stressed in a preliminary version of the surveys of Juhasz and Monk. We give a detailed proof for one strong nonpartition theorem (1.1) and then give various strengthenings. We then use 1.10 to get the consequences (in 1.11 and 1.12).<\/jats:p>","DOI":"10.2307\/2274951","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T18:33:48Z","timestamp":1146940428000},"page":"21-31","source":"Crossref","is-referenced-by-count":9,"title":["Strong negative partition above the continuum"],"prefix":"10.1017","volume":"55","author":[{"given":"Saharon","family":"Shelah","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200026402_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-21543-2"},{"key":"S0022481200026402_ref011","doi-asserted-by":"publisher","DOI":"10.1007\/BF02392561"},{"key":"S0022481200026402_ref013","first-page":"357","article-title":"Remarks on cellularity in products","volume":"57","author":"Todor\u010devi\u0107","year":"1986","journal-title":"Compositio Mathematica"},{"key":"S0022481200026402_ref007","volume-title":"Acta Mathematica Hungarica","author":"Shelah"},{"key":"S0022481200026402_ref008","unstructured":"Shelah S. , \u2135 \u03c9+1 has a J\u00f3nsson algebra (to appear)."},{"key":"S0022481200026402_ref006","doi-asserted-by":"publisher","DOI":"10.1007\/BF02787123"},{"key":"S0022481200026402_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF02760829"},{"key":"S0022481200026402_ref004","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(88)90052-8"},{"key":"S0022481200026402_ref005","doi-asserted-by":"publisher","DOI":"10.1007\/BF02783304"},{"key":"S0022481200026402_ref001","doi-asserted-by":"publisher","DOI":"10.1016\/B978-0-444-86580-9.50010-0"},{"key":"S0022481200026402_ref009","unstructured":"Shelah S. , There are J\u00f3nsson algebras in many inaccessibles (to appear)."},{"key":"S0022481200026402_ref012","first-page":"295","article-title":"Remark on chain conditions in products","volume":"55","author":"Todor\u010devi\u0107","year":"1985","journal-title":"Compositio Mathematica"},{"key":"S0022481200026402_ref010","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(87)90041-8"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200026402","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,18]],"date-time":"2019-05-18T17:52:39Z","timestamp":1558201959000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200026402\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1990,3]]},"references-count":13,"journal-issue":{"issue":"1","published-print":{"date-parts":[[1990,3]]}},"alternative-id":["S0022481200026402"],"URL":"https:\/\/doi.org\/10.2307\/2274951","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1990,3]]}}}