The universality spectrum of stable unsuperstable theories

by Kojman and Shelah. [KjSh:447]
Annals Pure and Applied Logic, 1992
It is shown that if T is stable unsuperstable, and aleph_1< lambda =cf(lambda)< 2^{aleph_0}, or 2^{aleph_0} < mu^+< lambda =cf(lambda)< mu^{aleph_0} then T has no universal model in cardinality lambda, and if e.g. aleph_omega < 2^{aleph_0} then T has no universal model in aleph_omega . These results are generalized to kappa =cf(kappa) < kappa (T) in the place of aleph_0 . Also: if there is a universal model in lambda >|T|, T stable and kappa < kappa (T) then there is a universal tree of height kappa +1 in cardinality lambda .

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