{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T22:11:44Z","timestamp":1648678304097},"reference-count":11,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2014,3,12]],"date-time":"2014-03-12T00:00:00Z","timestamp":1394582400000},"content-version":"unspecified","delay-in-days":5398,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1999,6]]},"abstract":"<jats:p>One of the early partition relation theorems which include ordinals was the observation of Erd\u00f6s and Rado [7] that if \u03ba = cf(\u03ba) &gt; \u03c9 then the Dushnik\u2013Miller theorem can be sharpened to \u03ba\u2192(\u03ba, \u03c9 + 1)<jats:sup>2<\/jats:sup>. The question on the possible further extension of this result was answered by Hajnal who in [8] proved that the continuum hypothesis implies \u03c9<jats:sub>1<\/jats:sub> \u219b (\u03c9<jats:sub>1<\/jats:sub>, \u03c9 + 2)<jats:sup>2<\/jats:sup>. He actually proved the stronger result \u03c9<jats:sub>1<\/jats:sub> \u219b (\u03c9: 2))<jats:sup>2<\/jats:sup>. The consistency of the relation \u03ba\u219b(\u03ba, (\u03c9: 2))<jats:sup>2<\/jats:sup> was later extensively studied. Baumgartner [1] proved it for every \u03ba which is the successor of a regular cardinal. Laver [9] showed that if \u03ba is Mahlo there is a forcing notion which adds a witness for \u03ba\u219b (\u03ba, (\u03c9: 2))<jats:sup>2<\/jats:sup> and preserves Mahloness, \u03c9-Mahloness of \u03ba, etc. We notice in connection with these results that \u03bb\u2192(\u03bb, (\u03c9: 2))<jats:sup>2<\/jats:sup> holds if \u03bb is singular, in fact \u03bb\u2192(\u03bb, (\u03bc: <jats:italic>n<\/jats:italic>))<jats:sup>2<\/jats:sup> for <jats:italic>n<\/jats:italic> &lt; \u03c9, \u03bc &lt; \u03bb (Theorem 4).<\/jats:p><jats:p>In [11] Todor\u010devi\u0107 proved that if cf(\u03bb) &gt; \u03c9 then a ccc forcing can add a counter-example to \u03bb\u2192(\u03bb, \u03c9 + 2)<jats:sup>2<\/jats:sup>. We give an alternative proof of this (Theorem 5) and extend it to larger cardinals: if GCH holds, cf (\u03bb) &gt; \u03ba = cf (\u03ba) then &lt; \u03ba-closed, \u03ba<jats:sup>+<\/jats:sup>-c.c. forcing adds a counter-example to \u03bb\u2192(\u03bb, \u03ba + 2)<jats:sup>2<\/jats:sup> (Theorem 6).<\/jats:p><jats:p>Erd\u00f6s and Hajnal remarked in their problem paper [5] that Galvin had proved \u03c9<jats:sub>2<\/jats:sub>\u2192(\u03c9<jats:sub>1<\/jats:sub>, \u03c9 + 2)<jats:sup>2<\/jats:sup> and he had also asked if \u03c9<jats:sub>2<\/jats:sub>\u2192(\u03c9<jats:sub>1<\/jats:sub>, \u03c9 + 3)<jats:sup>2<\/jats:sup> is true. We show in Theorem 1 that the negative relation is consistent.<\/jats:p>","DOI":"10.2307\/2586476","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T14:03:01Z","timestamp":1146924181000},"page":"436-442","source":"Crossref","is-referenced-by-count":1,"title":["Some remarks on the partition calculus of ordinals"],"prefix":"10.1017","volume":"64","author":[{"given":"P\u00e9ter","family":"Komj\u00e1th","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200013384_ref007","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1956-10036-0"},{"key":"S0022481200013384_ref004","first-page":"17\u201348","volume-title":"Proceedings of symposia in pure mathematics XIII","author":"Erd\u00f6s","year":"1971"},{"key":"S0022481200013384_ref006","volume-title":"Combinatorial set theory: Partition relation for cardinals","author":"Erd\u00f6s","year":"1984"},{"key":"S0022481200013384_ref009","first-page":"1029\u20131042","article-title":"Partition relations for uncountable cardinals <","volume":"10","author":"Laver","year":"1973","journal-title":"Infinite and finite sets, Keszthely, Hungary"},{"key":"S0022481200013384_ref005","first-page":"269\u2013287","volume-title":"Proceedings of the Tarski Symposium (Berkeley, California 1971)","author":"Erd\u00f6s","year":"1974"},{"key":"S0022481200013384_ref002","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(87)90077-7"},{"key":"S0022481200013384_ref010","doi-asserted-by":"publisher","DOI":"10.2307\/1999642"},{"key":"S0022481200013384_ref008","first-page":"277\u2013298","article-title":"Some results and problems in set theory","volume":"11","author":"Hajnal","year":"1960","journal-title":"Acta Mathematica Academlae Scientifica Hungarica"},{"key":"S0022481200013384_ref011","first-page":"159\u2013169","volume-title":"Logic, methodology, and philosophy of science, VII, Salzburg, 1983","volume":"114","author":"Todor\u010devi\u0107","year":"1986"},{"key":"S0022481200013384_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1940-07213-1"},{"key":"S0022481200013384_ref001","first-page":"401\u2013439","article-title":"Almost-disjoint sets, the dense set problem, and the partition calculus","volume":"10","author":"Baumgartner","year":"1976","journal-title":"Annals of Mathematical Logic"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200013384","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,9]],"date-time":"2019-05-09T17:18:29Z","timestamp":1557422309000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200013384\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,6]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1999,6]]}},"alternative-id":["S0022481200013384"],"URL":"https:\/\/doi.org\/10.2307\/2586476","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1999,6]]}}}