We prove two results on the stability spectrum for
L_{omega_1, omega} . Here S^m_i(M) denotes an appropriate
notion (at (+ 1 962) or mod) of Stone space of
m-types over M .
Theorem A. Suppose that for some positive integer m and
for every alpha < delta (T), there is an M in
boldmath K with
|S^m_i(M)| > |M|^{beth_alpha (|T|)} .
Then for every lambda >= |T|, there is an M with
|S^m_i(M)| > |M| .
Theorem B. Suppose that for every alpha < delta (T), there
is M_alpha in mbox {boldmath K} such that lambda_alpha =
|M_{alpha}| >= beth_alpha and |S^m_{i}(M_alpha)| >
lambda_alpha . Then for any mu with
mu^{aleph_0}> mu, mbox {boldmath 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 the Section~ ref {treeindis}, we expound the construction of
tree indiscernibles for sentences of L_{omega_1, omega} .
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