The Stability spectrum for classes of atomic models

We prove two results on the stability spectrum for L-omega 1,L-omega. Here S-i(m)(M) 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 alpha < delta(T), there is an M epsilon K with vertical bar S-i(m)(M)vertical bar > vertical bar M vertical bar(beth alpha(vertical bar T vertical bar)). Then for every lambda >= vertical bar T vertical bar, there is an M with vertical bar S-i(m) (M)vertical bar > vertical bar M vertical bar = lambda (2) Theorem for strictly stable case: Suppose that for every alpha < delta(T), there is M-alpha epsilon K such that lambda(alpha) = vertical bar M-alpha vertical bar >= beth(alpha) and vertical bar S-i(m) (M-alpha)vertical bar > lambda(alpha). Then for any mu with mu(N0) > mu, K is not i-stable in mu. 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-omega 1,L-omega. Further we provide some context for a number of variants on the Ehrenfeucht-Mostowski construction