{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T22:33:01Z","timestamp":1648679581425},"reference-count":23,"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":[[2012,6]]},"abstract":"<jats:p> We prove two results on the stability spectrum for L<jats:sub>\u03c9<jats:sub>1<\/jats:sub>,\u03c9<\/jats:sub>. Here [Formula: see text] denotes an appropriate notion (at or mod) of Stone space of m-types over M. (1) Theorem for unstable case: Suppose that for some positive integer m and for every \u03b1 &lt; \u03b4(T), there is an M \u2208 K with [Formula: see text]. Then for every \u03bb \u2265 |T|, there is an M with [Formula: see text]. (2) Theorem for strictly stable case: Suppose that for every \u03b1 &lt; \u03b4 (T), there is M<jats:sub>\u03b1<\/jats:sub> \u2208 K such that \u03bb<jats:sub>\u03b1<\/jats:sub> = |M<jats:sub>\u03b1<\/jats:sub>| \u2265 \u2136<jats:sub>\u03b1<\/jats:sub> and [Formula: see text]. Then for any \u03bc with \u03bc<jats:sup>\u2135<jats:sub>0<\/jats:sub><\/jats:sup> &gt; \u03bc, K is not i-stable in \u03bc. These results provide a new kind of sufficient condition for the unstable case and shed some light on the spectrum of strictly stable theories in this context. The methods avoid the use of compactness in the theory under study. In this paper, we expound the construction of tree indiscernibles for sentences of L<jats:sub>\u03c9<jats:sub>1<\/jats:sub>,\u03c9<\/jats:sub>. Further we provide some context for a number of variants on the Ehrenfeucht\u2013Mostowski construction. <\/jats:p>","DOI":"10.1142\/s0219061312500018","type":"journal-article","created":{"date-parts":[[2012,4,20]],"date-time":"2012-04-20T11:45:27Z","timestamp":1334922327000},"page":"1250001","source":"Crossref","is-referenced-by-count":3,"title":["THE STABILITY SPECTRUM FOR CLASSES OF ATOMIC MODELS"],"prefix":"10.1142","volume":"12","author":[{"given":"JOHN T.","family":"BALDWIN","sequence":"first","affiliation":[{"name":"Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. 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