{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T10:54:23Z","timestamp":1777373663290,"version":"3.51.4"},"reference-count":18,"publisher":"World Scientific Pub Co Pte Lt","issue":"01","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Math. Log."],"published-print":{"date-parts":[[2006,6]]},"abstract":"<jats:p> We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper, we explore stability results in this new context. We assume that [Formula: see text] is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include:. <\/jats:p><jats:p> Theorem 0.1. Suppose that [Formula: see text] is not only tame, but [Formula: see text]-tame. If [Formula: see text] and [Formula: see text] is Galois stable in \u03bc, then [Formula: see text], where [Formula: see text] is a relative of \u03ba(T) from first order logic. <\/jats:p><jats:p> [Formula: see text] is the Hanf number of the class [Formula: see text]. It is known that [Formula: see text]. <\/jats:p><jats:p> The theorem generalizes a result from [17]. It is used to prove both the existence of Morley sequences for non-splitting (improving [22, Claim 4.15] and a result from [7]) and the following initial step towards a stability spectrum theorem for tame classes:. <\/jats:p><jats:p> Theorem 0.2. If [Formula: see text] is Galois-stable in some [Formula: see text], then [Formula: see text] is stable in every \u03ba with \u03ba<jats:sup>\u03bc<\/jats:sup>=\u03ba. For example, under GCH we have that [Formula: see text] Galois-stable in \u03bc implies that [Formula: see text] is Galois-stable in \u03bc<jats:sup>+n<\/jats:sup> for all n &lt; \u03c9. <\/jats:p>","DOI":"10.1142\/s0219061306000487","type":"journal-article","created":{"date-parts":[[2006,7,24]],"date-time":"2006-07-24T09:26:26Z","timestamp":1153733186000},"page":"25-48","source":"Crossref","is-referenced-by-count":62,"title":["GALOIS-STABILITY FOR TAME ABSTRACT ELEMENTARY CLASSES"],"prefix":"10.1142","volume":"06","author":[{"given":"RAMI","family":"GROSSBERG","sequence":"first","affiliation":[{"name":"Department of Mathematics, Carnegie Mellon University, Pittsburgh PA 15213, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MONICA","family":"VANDIEREN","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Michigan, Ann Arbor MI 48109-1109, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,21]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.2307\/2274488"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/302\/05080"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/s001530050157"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1007\/s001530200000"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.2178\/jsl\/1146620158"},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1975-0387043-7"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02762619"},{"key":"rf15","first-page":"41","volume":"47","author":"Makkai M.","journal-title":"Ann. 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