The stability spectrum for classes of atomic models

We prove two results on the stability spectrum for L _ω1,ω. Here S ^m _i(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 α < δ(T), there is an M ∈ K with | S ^m _i(M)| > |M|α ^(|T|). Then for every λ < |T|, there is an M with |S ^m _i(M)| > |M| = λ. (2) Theorem for strictly stable case: Suppose that for every α < δ(T), there is M _α ∈ K such that λ _α = |M _α|< _α and |S ^m _i(M _α)| > λ _α. Then for any μ with μ ^α ₀ > μ, K is not i-stable in μ. 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 _ω1,ω. Further we provide some context for a number of variants on the EhrenfeuchtMostowski construction.