{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,28]],"date-time":"2022-03-28T23:07:08Z","timestamp":1648508828721},"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":7497,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1993,9]]},"abstract":"<jats:p>We will consider some partition properties of the following type: given a function <jats:italic>F<\/jats:italic>: \u03c9<jats:sup>\u03c9<\/jats:sup> \u21922, is there a sequence <jats:italic>H<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>H<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026 of subsets of \u03c9 such that <jats:italic>F<\/jats:italic> is constant on \u03a0<jats:sub><jats:italic>i<\/jats:italic>\u03b5\u03c9<\/jats:sub><jats:italic>H<\/jats:italic><jats:sub>i<\/jats:sub>? The answer is obviously positive if we allow all the H<jats:sub>i<\/jats:sub>'s to have exactly one element, but the problem is nontrivial if we require the H<jats:sub>i<\/jats:sub>'s to have at least two elements. The axiom of choice contradicts the statement \u201cfor all <jats:italic>F<\/jats:italic>: \u03c9<jats:sup>\u03c9<\/jats:sup>\u2192 2 there is a sequence <jats:italic>H<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>H<\/jats:italic><jats:sub>1<\/jats:sub>, <jats:italic>H<\/jats:italic><jats:sub>2<\/jats:sub>,\u2026 of subsets of \u03c9 such that {<jats:italic>i<\/jats:italic>|(H<jats:sub>i<\/jats:sub>) \u2265 2} is infinite and <jats:italic>F<\/jats:italic> is constant on \u03a0<jats:italic>H<\/jats:italic><jats:sub>i<\/jats:sub>\u201d, but the infinite exponent partition relation \u03c9(\u03c9)<jats:sup>\u03c9<\/jats:sup> implies it; so, this statement is relatively consistent with an inaccessible cardinal. (See [1] where these partition properties were considered.)<\/jats:p><jats:p>We will also consider partitions into any finite number of pieces, and we will prove some facts about partitions into \u03c9-many pieces.<\/jats:p><jats:p>Given a partition <jats:italic>F<\/jats:italic>: \u03c9<jats:sup>\u03c9<\/jats:sup> \u2192 <jats:italic>k<\/jats:italic>, we say that <jats:italic>H<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>H<\/jats:italic><jats:sub>1<\/jats:sub>\u2026, a sequence of subsets of \u03c9, is homogeneous for <jats:italic>F<\/jats:italic> if <jats:italic>F<\/jats:italic> is constant on \u03a0<jats:italic>H<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>. We say the sequence <jats:italic>H<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>H<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026 is nonoverlapping if, for all <jats:italic>i<\/jats:italic> \u2208 \u03c9, \u222a<jats:italic>H<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> &gt; \u2229<jats:italic>H<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic>+1<\/jats:sub>.<\/jats:p><jats:p>The sequence \u3008<jats:italic>H<\/jats:italic><jats:sub>i<\/jats:sub>: <jats:italic>i<\/jats:italic> \u2208 \u03c9\u3009 is of type \u3008\u03b1<jats:sub>0<\/jats:sub>, \u03b1<jats:sub>1<\/jats:sub>,\u2026\u3009 if, for every <jats:italic>i<\/jats:italic> \u2208 \u03c9, \u2223H<jats:sub>i<\/jats:sub>\u2223 = \u03b1<jats:sub>i<\/jats:sub>.<\/jats:p><jats:p>We will adopt the usual notation for polarized partition relations due to Erd\u00f6s, Hajnal, and Rado.<\/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=\"S0022481200020922_eqnU1\" \/><\/jats:disp-formula><\/jats:p><jats:p>means that for every partition <jats:italic>F<\/jats:italic>: \u03ba<jats:sub>1<\/jats:sub> \u00d7 \u03ba<jats:sub>2<\/jats:sub> \u00d7 \u2026 \u00d7 \u03ba<jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>\u2192\u03b4 there is a sequence <jats:italic>H<\/jats:italic><jats:sub>0<\/jats:sub>, <jats:italic>H<\/jats:italic><jats:sub>1<\/jats:sub>,\u2026, <jats:italic>H<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub> such that <jats:italic>H<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> \u2282 \u03ba<jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> and \u2223<jats:italic>H<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\u2223 = \u03b1<jats:sub><jats:italic>i<\/jats:italic><\/jats:sub> for every <jats:italic>i<\/jats:italic>, 1 \u2264 <jats:italic>i<\/jats:italic> \u2264 <jats:italic>n<\/jats:italic>, and <jats:italic>F<\/jats:italic> is constant on H<jats:sub>1<\/jats:sub> \u00d7 <jats:italic>H<\/jats:italic><jats:sub>2<\/jats:sub> \u00d7 \u2026 \u00d7 <jats:italic>H<\/jats:italic><jats:sub><jats:italic>n<\/jats:italic><\/jats:sub>.<\/jats:p>","DOI":"10.2307\/2275100","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T22:49:16Z","timestamp":1146955756000},"page":"860-871","source":"Crossref","is-referenced-by-count":11,"title":["Partitions of products"],"prefix":"10.1017","volume":"58","author":[{"given":"Carlos A.","family":"Di Prisco","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"James M.","family":"Henle","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200020922_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1956-10036-0"},{"key":"S0022481200020922_ref004","doi-asserted-by":"publisher","DOI":"10.1112\/S002557930001010X"},{"key":"S0022481200020922_ref001","doi-asserted-by":"publisher","DOI":"10.1002\/malq.19930390150"},{"key":"S0022481200020922_ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF01886396"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200020922","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,15]],"date-time":"2019-05-15T22:02:37Z","timestamp":1557957757000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200020922\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1993,9]]},"references-count":4,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1993,9]]}},"alternative-id":["S0022481200020922"],"URL":"https:\/\/doi.org\/10.2307\/2275100","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1993,9]]}}}