{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,5]],"date-time":"2022-04-05T14:02:32Z","timestamp":1649167352154},"reference-count":6,"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":15076,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1972,12]]},"abstract":"<jats:p>Although there are many characterizations of weakly compact cardinals (e.g. in terms of indescnbability and tree properties as well as compactness) the most interesting set-theoretic (combinatorial) one is in terms of partition relations. To be more precise we define for <jats:italic>\u03ba<\/jats:italic> and <jats:italic>\u03b1<\/jats:italic> cardinals and <jats:italic>n<\/jats:italic> an integer the partition relation <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200078476_inline1\" \/> of Erd\u00f6s, Hajnal and Rado [2] as follows:<\/jats:p><jats:p>For every function <jats:italic>F<\/jats:italic>: [<jats:italic>\u03ba<\/jats:italic>]<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup><jats:italic>\u2192 \u03b1<\/jats:italic> (called a partition of [<jats:italic>\u03ba<\/jats:italic>]<jats:sup><jats:italic>n<\/jats:italic><\/jats:sup>, the <jats:italic>n<\/jats:italic>-element subsets of <jats:italic>\u03ba<\/jats:italic>, into \u03b1 pieces), there exists a set <jats:italic>C<\/jats:italic><jats:italic>\u2286 \u03ba<\/jats:italic> (called homogeneous for <jats:italic>F<\/jats:italic>) such that card <jats:italic>C<\/jats:italic> = <jats:italic>\u03ba<\/jats:italic> and <jats:italic>F<\/jats:italic>\u2033[<jats:italic>C<\/jats:italic>]<jats:italic>n<\/jats:italic><jats:italic>\u2260 \u03b1<\/jats:italic>, i.e. some element of the range is omitted when <jats:italic>F<\/jats:italic> is restricted to the <jats:italic>n<\/jats:italic>-element subsets of <jats:italic>C<\/jats:italic>. It is the simplest (nontrivial) of these relations, i.e. <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200078476_inline2\" \/>, that is the well-known equivalent of weak compactness.<jats:sup>1<\/jats:sup><\/jats:p><jats:p>Two directions of inquiry immediately suggest themselves when weak compactness is described in terms of these partition relations: (a) Trying to strengthen the relation by increasing the superscript\u2014e.g., <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200078476_inline3\" \/>\u2014and (b) trying to weaken the relation by increasing the subscript\u2014e.g., <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200078476_inline4\" \/>. As it turns out, the strengthening to <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200078476_inline3\" \/> is only illusory for using the equivalence of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200078476_inline2\" \/> to the tree property one quickly sees that <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200078476_inline2\" \/> implies (and so is equivalent to) <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200078476_inline5\" \/> for every <jats:italic>n<\/jats:italic>. Thus <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200078476_inline2\" \/> is the strongest of these partition relations. The second question seems much more difficult.<\/jats:p>","DOI":"10.2307\/2272412","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T21:19:43Z","timestamp":1146950383000},"page":"673-676","source":"Crossref","is-referenced-by-count":3,"title":["Weak compactness and square bracket partition relations"],"prefix":"10.1017","volume":"37","author":[{"given":"E. M.","family":"Kleinberg","sequence":"first","affiliation":[]},{"given":"R. A.","family":"Shore","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200078476_ref003","first-page":"666","article-title":"Negation of partition relations without CH","volume":"18","author":"Galvin","year":"1971","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200078476_ref005","unstructured":"Shore R. A. , Square bracket partition relations in L (to appear)."},{"key":"S0022481200078476_ref002","doi-asserted-by":"publisher","DOI":"10.1007\/BF01886396"},{"key":"S0022481200078476_ref001","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1972-0297576-7"},{"key":"S0022481200078476_ref004","first-page":"296","article-title":"Somewhat homogeneous sets. III","volume":"17","author":"Kleinberg","year":"1970","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200078476_ref006","unstructured":"Silver J. , Some applications of model theory in set theory. Doctoral Dissertation, University of California, Berkeley, Calif., 1966."}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200078476","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T20:32:59Z","timestamp":1559248379000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200078476\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1972,12]]},"references-count":6,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1972,12]]}},"alternative-id":["S0022481200078476"],"URL":"https:\/\/doi.org\/10.2307\/2272412","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1972,12]]}}}