{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,3]],"date-time":"2022-04-03T00:01:41Z","timestamp":1648944101767},"reference-count":16,"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":4210,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. symb. log."],"published-print":{"date-parts":[[2002,9]]},"abstract":"<jats:p>Various results have been proved about growth rates of certain sequences of integers associated with infinite permutation groups. Most of these concern the number of orbits of the automorphism group of an \u2135<jats:sub>0<\/jats:sub>-categorical structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/> on the set of unordered <jats:italic>n<\/jats:italic>-subsets or on the set of <jats:italic>n<\/jats:italic>-tuples of elements of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/>. (Recall that by the Ryll-Nardzewski Theorem, if <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/> is countable and \u2135<jats:sub>0<\/jats:sub>-categorical, the number of the orbits of its automorphism group Aut(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/>) on the set of <jats:italic>n<\/jats:italic>-tuples from <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/> is finite and equals the number of complete <jats:italic>n<\/jats:italic>-types consistent with the theory of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/>.) The book [Ca90] is a convenient reference for these results. One of the oldest (in the realms of \u2018folklore\u2019) is that for any sequence (<jats:italic>K<jats:sub>n<\/jats:sub><\/jats:italic>)<jats:sub><jats:italic>n<\/jats:italic>\u2208\u2115<\/jats:sub> of natural numbers there is a countable \u2135<jats:sub>0<\/jats:sub>-categorical structure <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/> such that the number of orbits of Aut(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/>) on the set of <jats:italic>n<\/jats:italic>-tuples from <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/> is greater than <jats:italic>k<jats:sub>n<\/jats:sub><\/jats:italic> for all <jats:italic>n<\/jats:italic>.<\/jats:p><jats:p>These investigations suggested the study of the growth rate of another sequence. Let <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/> be an \u2135<jats:sub>0<\/jats:sub>-categorical structure and <jats:italic>X<\/jats:italic> be a finite subset of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/>. Let acl(<jats:italic>X<\/jats:italic>) be the algebraic closure of <jats:italic>X<\/jats:italic>, that is, the union of the finite <jats:italic>X<\/jats:italic>-definable subsets of <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/>. Equivalently, this is the union of the finite orbits on <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/> of Aut(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/>)(<jats:sub><jats:italic>X<\/jats:italic><\/jats:sub>), the pointwise stabiliser of <jats:italic>X<\/jats:italic> in Aut(<jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200009300_inline1\" \/>). Define<\/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=\"S0022481200009300_eqnU1\" \/><\/jats:disp-formula><\/jats:p>","DOI":"10.2178\/jsl\/1190150137","type":"journal-article","created":{"date-parts":[[2007,12,13]],"date-time":"2007-12-13T19:14:26Z","timestamp":1197573266000},"page":"897-909","source":"Crossref","is-referenced-by-count":0,"title":["\u2135<sub>0<\/sub>-categorical structures with arbitrarily fast growth of algebraic closure"],"prefix":"10.1017","volume":"67","author":[{"given":"David M.","family":"Evans","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M. E.","family":"Pantano","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2014,3,12]]},"reference":[{"key":"S0022481200009300_ref014","unstructured":"Pantano M.E. , Algebraic closure in \u21350-categorical structures, Ph.D. thesis , University of East Anglia, Norwich, 1995."},{"key":"S0022481200009300_ref013","doi-asserted-by":"publisher","DOI":"10.1093\/qmath\/37.4.449"},{"key":"S0022481200009300_ref015","doi-asserted-by":"publisher","DOI":"10.1007\/BF02773695"},{"key":"S0022481200009300_ref004","volume-title":"Automorphisms of first-order structures","author":"Evans","year":"1994"},{"key":"S0022481200009300_ref003","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(85)90023-5"},{"key":"S0022481200009300_ref011","unstructured":"Hrushovski Ehud , Simplicity and the Lascar group, Preprint, 1997."},{"key":"S0022481200009300_ref005","volume-title":"Annals of Pure and Applied Logic","author":"Evans","year":"2000"},{"key":"S0022481200009300_ref007","doi-asserted-by":"publisher","DOI":"10.24033\/asens.1027"},{"key":"S0022481200009300_ref002","volume-title":"Model theory","author":"Chang","year":"1990"},{"key":"S0022481200009300_ref001","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511549809"},{"key":"S0022481200009300_ref009","unstructured":"Hrushovski Ehud , A stable \u21350-categorical pseudoplane, Unpublished notes, 1988."},{"key":"S0022481200009300_ref010","doi-asserted-by":"publisher","DOI":"10.1016\/0168-0072(93)90171-9"},{"key":"S0022481200009300_ref016","volume-title":"Automorphisms of first-order structures","author":"Wagner","year":"1994"},{"key":"S0022481200009300_ref008","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511551574"},{"key":"S0022481200009300_ref006","first-page":"540","article-title":"Sur certains relations qui g\u00e9n\u00e9ralised l'ordre des nombres rationnels","volume":"237","author":"Fra\u00efss\u00e9","year":"1953","journal-title":"Comptes Rendus de l'Acad\u00e9mie des Sciences, Paris"},{"key":"S0022481200009300_ref012","doi-asserted-by":"publisher","DOI":"10.1305\/ndjfl\/1093636094"}],"container-title":["Journal of Symbolic Logic"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0022481200009300","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,7]],"date-time":"2019-05-07T00:45:21Z","timestamp":1557189921000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0022481200009300\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,9]]},"references-count":16,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2002,9]]}},"alternative-id":["S0022481200009300"],"URL":"https:\/\/doi.org\/10.2178\/jsl\/1190150137","relation":{},"ISSN":["0022-4812","1943-5886"],"issn-type":[{"value":"0022-4812","type":"print"},{"value":"1943-5886","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,9]]}}}