The Stability spectrum for classes of atomic models

We prove two results on the stability spectrum for Lω1,ω . Here S m i (M) denotes an appropriate notion (at or mod) of Stone space of m-types over M . Theorem A. Suppose that for some positive integer m and for every α < δ(T ), there is an M ∈ K with |S i (M)| > |M |iα(|T |). Then for every λ ≥ |T |, there is an M with |S i (M)| > |M |. Theorem B. Suppose that for every α < δ(T ), there is Mα ∈ K such that λα = |Mα| ≥ iα and |S i (Mα)| > λα. Then for any μ with μא0 > 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 the Section 4, we expound the construction of tree indiscernibles for sentences of Lω1,ω . Further we provide some context for a number of variants on the Ehrenfeucht-Mostowski construction.