Abstract
Let K be an abstract elementary class of models. Assume that there are less than the maximal number of models in K_{λ^{+n}} (namely models in K of power λ^{+n}) for all n. We provide conditions on K_λ, that imply the existence of a model in K_{λ^{+n}} for all n. We do this by providing sufficiently strong conditions on K_λ, that they are inherited by a properly chosen subclass of K_{λ^+}.
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