Kojman, M. and S. Shelah, The universality spectrum of stable unsuperstable theories, Annals of Pure and Applied Logic 58 (1992) 57-72. It is shown that if T is stable unsuperstable, and א1<λ = cf λ<2א0, or 2א0<μ+<λ = cf λ<μא0 then T has no universal model in cardinality λ, and if e.g. אω<2א0 then T has no universal model in אω. These results are generalized to κ = cf κ<κ(T) in place of א0. Also: if there is a universal model in λ>[brvbar]T[brvbar], T stable and κ<κ(T) then there is a universal tree of height κ + 1 in cardinality λ.
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