{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T18:01:48Z","timestamp":1775844108628,"version":"3.50.1"},"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":14894,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[1973,6]]},"abstract":"Definition 1. For a set S<\/jats:italic> and a cardinal \u03ba<\/jats:italic>,<\/jats:p><\/jats:disp-formula><\/jats:p>In particular, 2\u03c9<\/jats:sup> denotes the power set of the natural numbers and not the cardinal 2\u21350<\/jats:sup>. We regard 2\u03c9<\/jats:sup> as a topological space with the usual product topology.<\/jats:p>Definition 2. A set S<\/jats:italic> \u2286 2\u03c9<\/jats:sup> is Ramsey<\/jats:italic> if there is an M<\/jats:italic> \u2208 [\u03c9]\u03c9<\/jats:sup> such that either [M<\/jats:italic>]\u03c9<\/jats:sup> \u2286 S<\/jats:italic> or else [M<\/jats:italic>]\u03c9<\/jats:sup> \u2286 2\u03c9<\/jats:sup> \u2212 S<\/jats:italic>.<\/jats:p>Erd\u00f6s and Rado [3, Example 1, p. 434] showed that not every S<\/jats:italic> \u2286 2\u03c9<\/jats:sup> is Ramsey. In view of the nonconstructive character of the counterexample, one might expect (as Dana Scott has suggested) that all sufficiently definable sets are Ramsey. In fact, our main result (Theorem 2) is that all Borei sets are Ramsey. Soare [10] has applied this result to some problems in recursion theory.<\/jats:p>The first positive result on Scott's problem was Ramsey's theorem [8, Theorem A]. The next advance was Nash-Williams' generalization of Ramsey's theorem (Corollary 2), which can be interpreted as saying: If S<\/jats:italic>1<\/jats:sub> and S<\/jats:italic>2<\/jats:sub> are disjoint open subsets of 2\u03c9<\/jats:sup>, there is an M<\/jats:italic> \u2208 [\u03c9]\u03c9<\/jats:sup> such that either [M<\/jats:italic>]\u03c9<\/jats:sup> \u22c2 S<\/jats:italic>1<\/jats:sub> = \u2205 or [M<\/jats:italic>]\u03c9<\/jats:sup> \u2229 S<\/jats:italic>2<\/jats:sub> = \u2286. (This is halfway between \u201cclopen sets are Ramsey\u201d and \u201copen sets are Ramsey.\u201d) Then Galvin [4] stated a generalization of Nash-Williams' theorem (Corollary 1) which says, in effect, that open sets are Ramsey; this was discovered independently by Andrzej Ehrenfeucht, Paul Cohen, and probably many others, but no proof has been published.<\/jats:p>","DOI":"10.2307\/2272055","type":"journal-article","created":{"date-parts":[[2006,5,6]],"date-time":"2006-05-06T17:22:43Z","timestamp":1146936163000},"page":"193-198","source":"Crossref","is-referenced-by-count":148,"title":["Borel sets and Ramsey's theorem"],"prefix":"10.1017","volume":"38","author":[{"given":"Fred","family":"Galvin","sequence":"first","affiliation":[]},{"given":"Karel","family":"Prikry","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200077793_ref004","first-page":"548","article-title":"A generalization of Ramsey's theorem","volume":"15","author":"Galvin","year":"1968","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200077793_ref009","first-page":"60","volume":"35","author":"Silver","year":"1970","journal-title":"Every analytic set is Ramsey"},{"key":"S0022481200077793_ref003","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-2.1.417"},{"key":"S0022481200077793_ref005","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90009-4"},{"key":"S0022481200077793_ref001","unstructured":"Chang C. C. and Galvin Fred , A combinatorial theorem with applications to polarized partition relations (to appear)."},{"key":"S0022481200077793_ref002","first-page":"17","volume-title":"Proceedings of Symposia in Pure Mathematics","volume":"13","author":"Erd\u00f6s","year":"1971"},{"key":"S0022481200077793_ref008","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s2-30.1.264"},{"key":"S0022481200077793_ref007","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100038603"},{"key":"S0022481200077793_ref011","doi-asserted-by":"publisher","DOI":"10.2307\/1970696"},{"key":"S0022481200077793_ref006","first-page":"931","article-title":"On a generalization of Ramsey's theorem","volume":"15","author":"Mathias","year":"1968","journal-title":"Notices of the American Mathematical Society"},{"key":"S0022481200077793_ref010","first-page":"53","volume":"34","author":"Soare","year":"1969","journal-title":"Sets with no subset of higher degree"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200077793","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T15:54:16Z","timestamp":1559231656000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200077793\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1973,6]]},"references-count":11,"journal-issue":{"issue":"2","published-print":{"date-parts":[[1973,6]]}},"alternative-id":["S0022481200077793"],"URL":"https:\/\/doi.org\/10.2307\/2272055","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[1973,6]]}}}