### Non forking good frames minus local character

by Jarden and Shelah. [JaSh:940]

Notre Dame J Formal Logic, 2013

We prove that if s is an almost good lambda-frame
(i.e. s is a good lambda-frame except that it satisfies
just a weak version of local character), then we can complete
the frame s.t. it will satisfy local character too. This
theorem has an important application. In [she46] it has
proved that (under mild set theoretic assumptions)
Categoricity in lambda, lambda^+ and intermediate number of
models in K_{lambda^{++}} implies existence of an almost
good lambda-frame. So by our theorem, we can get the local
character too. So by categoricity assumptions in
lambda, lambda^+, lambda^{++} we can get existence of a good
lambda-frame. Combining this with [sh600], we conclude
that the function lambda-> I(lambda,K), which
correspond to each cardinal lambda, the number of models in K
of cardinality lambda, is not arbitrary.

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