{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T02:39:17Z","timestamp":1777516757911,"version":"3.51.4"},"reference-count":21,"publisher":"SAGE Publications","issue":"4","license":[{"start":{"date-parts":[[2016,9,6]],"date-time":"2016-09-06T00:00:00Z","timestamp":1473120000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Computability"],"published-print":{"date-parts":[[2017,10,31]]},"abstract":"<jats:p>Ramsey\u2019s theorem for pairs asserts that every 2-coloring of the pairs of integers has an infinite monochromatic subset. In this paper, we study a strengthening of Ramsey\u2019s theorem for pairs due to Erd\u0151s and Rado, which states that every 2-coloring of the pairs of rationals has either an infinite 0-homogeneous set or a 1-homogeneous set of order type \u03b7, where\u00a0 \u03b7 is the order type of the rationals. This theorem is a natural candidate to lie strictly between the arithmetic comprehension axiom and Ramsey\u2019s theorem for pairs. This Erd\u0151s\u2013Rado theorem, like the tree theorem for pairs, belongs to a family of Ramsey-type statements whose logical strength remains a challenge.<\/jats:p>","DOI":"10.3233\/com-160067","type":"journal-article","created":{"date-parts":[[2016,9,6]],"date-time":"2016-09-06T11:16:38Z","timestamp":1473160598000},"page":"319-331","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":4,"title":["Coloring the rationals in reverse mathematics"],"prefix":"10.1177","volume":"6","author":[{"given":"Emanuele","family":"Frittaion","sequence":"first","affiliation":[{"name":"Mathematical Institute, Tohoku University, Japan. \u00a0"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ludovic","family":"Patey","sequence":"additional","affiliation":[{"name":"Laboratoire PPS, Paris Diderot University, France. \u00a0"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2016,9,6]]},"reference":[{"key":"ref001","doi-asserted-by":"publisher","DOI":"10.2307\/2694910"},{"key":"ref002","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1231082309"},{"key":"ref003","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1278682209"},{"key":"ref004","doi-asserted-by":"publisher","DOI":"10.1007\/s00153-010-0179-6"},{"key":"ref005","doi-asserted-by":"publisher","DOI":"10.1142\/9208"},{"key":"ref006","unstructured":"D.D.\u00a0Dzhafarov, Strong reductions between combinatorial principles,\n                      Journal of Symbolic Logic\n                      , to appear."},{"key":"ref007","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-2.1.417"},{"key":"ref008","doi-asserted-by":"publisher","DOI":"10.1002\/malq.201200025"},{"key":"ref009","unstructured":"J.L.\u00a0Hirst, Combinatorics in subsystems of second order arithmetic, PhD thesis, Pennsylvania State University, August 1987."},{"key":"ref010","doi-asserted-by":"publisher","DOI":"10.2307\/2272972"},{"key":"ref011","first-page":"33","volume":"173","author":"Jockusch C.G.","year":"1972","journal-title":"Transactions of the American Mathematical Society"},{"key":"ref012","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061313500074"},{"key":"ref013","doi-asserted-by":"publisher","DOI":"10.2178\/bsl\/1309952320"},{"key":"ref014","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-20028-6_30"},{"key":"ref015","unstructured":"L.\u00a0Patey, The strength of the tree theorem for pairs in reverse mathematics.\n                      Journal of Symbolic Logic\n                      , to appear."},{"key":"ref016","unstructured":"L.\u00a0Patey, The weakness of being cohesive, thin or free in reverse mathematics,\n                      Israel Journal of Mathematics\n                      , to appear. Available at: http:\/\/arxiv.org\/abs\/1502.03709."},{"key":"ref017","unstructured":"J.G.\u00a0Rosenstein, Linear Orderings, Pure and Applied Mathematics, Vol.\u00a098, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982."},{"key":"ref018","doi-asserted-by":"publisher","DOI":"10.1090\/pspum\/005\/0141595"},{"key":"ref019","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1040136917"},{"key":"ref020","doi-asserted-by":"publisher","DOI":"10.2307\/1970028"},{"key":"ref021","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511581007"}],"container-title":["Computability"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/COM-160067","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/COM-160067","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/COM-160067","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T15:59:54Z","timestamp":1777391994000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/full\/10.3233\/COM-160067"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,9,6]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2017,10,31]]}},"alternative-id":["10.3233\/COM-160067"],"URL":"https:\/\/doi.org\/10.3233\/com-160067","relation":{},"ISSN":["2211-3568","2211-3576"],"issn-type":[{"value":"2211-3568","type":"print"},{"value":"2211-3576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,9,6]]}}}