Non forking good frames minus local character

by Jarden and Shelah. [JaSh:940]

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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