{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T13:53:59Z","timestamp":1773237239123,"version":"3.50.1"},"reference-count":17,"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":1927,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2008,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We show that if add(null) = c, then the globally Baire and universally measurable chromatic numbers of the graph of any Borel function on a Polish space are equal and at most three. In particular, this holds for the graph of the unilateral shift on [\u2115]<jats:sup>\u2115<\/jats:sup>, although its Borel chromatic number is \u2135<jats:sub>0<\/jats:sub>. We also show that if add(null) = c, then the universally measurable chromatic number of every treeing of a measure amenable equivalence relation is at most three. In particular, this holds for \u201cthe\u201d minimum analytic graph <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200003911_inline1\"\/> with uncountable Borel (and Baire measurable) chromatic number. In contrast, we show that for all \u03ba \u04046 (2, 3\u2026..\u2135<jats:sub>0<\/jats:sub>, c), there is a treeing of <jats:italic>E<\/jats:italic><jats:sub>0<\/jats:sub> with Borel and Baire measurable chromatic number \u03ba. Finally, we use a Glimm\u2013Effros style dichotomy theorem to show that every basis for a non-empty initial segment of the class of graphs of Borel functions of Borel chromatic number at least three contains a copy of (\u211d&lt;\u2115, \u2287).<\/jats:p>","DOI":"10.2178\/jsl\/1230396910","type":"journal-article","created":{"date-parts":[[2008,12,27]],"date-time":"2008-12-27T16:55:26Z","timestamp":1230396926000},"page":"1139-1157","source":"Crossref","is-referenced-by-count":4,"title":["Measurable chromatic numbers"],"prefix":"10.1017","volume":"73","author":[{"given":"Benjamin D.","family":"Miller","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200003911_ref015","volume-title":"Ergodic Theory of Dynamic Systems","author":"Miller"},{"key":"S0022481200003911_ref013","doi-asserted-by":"publisher","DOI":"10.1016\/0003-4843(70)90009-4"},{"key":"S0022481200003911_ref001","doi-asserted-by":"publisher","DOI":"10.2307\/1999530"},{"key":"S0022481200003911_ref012","unstructured":"Louveau A. , Some dichotomy results for analytic graphs, preprint, 2001."},{"key":"S0022481200003911_ref005","article-title":"Ends of graphed equivalence relations, II","author":"Hjorth","journal-title":"Israel Journal of Mathematics"},{"key":"S0022481200003911_ref003","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-98-04489-X"},{"key":"S0022481200003911_ref007","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4190-4"},{"key":"S0022481200003911_ref002","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1994-1149121-0"},{"key":"S0022481200003911_ref004","doi-asserted-by":"publisher","DOI":"10.1090\/S0894-0347-1990-1057041-5"},{"key":"S0022481200003911_ref010","doi-asserted-by":"publisher","DOI":"10.1006\/aima.1998.1771"},{"key":"S0022481200003911_ref008","doi-asserted-by":"publisher","DOI":"10.1007\/b99421"},{"key":"S0022481200003911_ref009","doi-asserted-by":"publisher","DOI":"10.1007\/s11856-008-0011-8"},{"key":"S0022481200003911_ref011","doi-asserted-by":"crossref","unstructured":"Lecomte D. and Miller B. D. , Basis theorems for non-potentially closed sets and graphs of uncountable Borel chromatic number, preprint, 2007.","DOI":"10.1142\/S0219061308000749"},{"key":"S0022481200003911_ref017","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-9964-7"},{"key":"S0022481200003911_ref006","doi-asserted-by":"publisher","DOI":"10.1142\/S0219061302000138"},{"key":"S0022481200003911_ref016","article-title":"Descriptive Kakutani equivalence","author":"Miller","journal-title":"Journal of the European Mathematical Society"},{"key":"S0022481200003911_ref014","article-title":"Ends of graphed equivalence relations. I","author":"Miller","journal-title":"Israel Journal of Mathematics"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200003911","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,4,30]],"date-time":"2019-04-30T21:35:52Z","timestamp":1556660152000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200003911\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,12]]},"references-count":17,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2008,12]]}},"alternative-id":["S0022481200003911"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1230396910","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,12]]}}}