The universality spectrum of stable unsuperstable theories

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<2^א0, or 2^א0<μ⁺<λ = cf λ<μ^א0 then T has no universal model in cardinality λ, and if e.g. א_ω<2^א0 then T has no universal model in א_ω. 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.