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Shelah has provided sufficient conditions for an ${\Bbb L}_{\omega _1 ,\omega } $ -sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ${\aleph _n}$ 's. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\aleph _0}$ and ${\aleph _1}$ which suffice when ψ is restricted to be universal: Theorem. Assume ${2^{{\aleph _0}}}. Let ψ be a universal ${\Bbb L}_{\omega _1 ,\omega } $ -sentence. (1) If ψ is categorical in ${\aleph _0}$ and $1 \leqslant {\Bbb L}\left({\psi ,\aleph _1 } \right) , then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals. (2) If ψ is categorical in ${\aleph _1}$ , then ψ is categorical in all uncountable cardinals. The theorem generalizes to the framework of ${\Bbb L}_{\omega _1 ,\omega } $ -definable tame abstract elementary classes with primes. [ABSTRACT FROM AUTHOR]

In the context of abstract elementary classes (AECs) with a monster model, several possible definitions of superstability have appeared in the literature. Among them are no long splitting chains, uniqueness of limit models, and solvability. Under the assumption that the class is tame and stable, we show that (asymptotically) no long splitting chains implies solvability and uniqueness of limit models implies no long splitting chains. Using known implications, we can then conclude that all the previously-mentioned definitions (and more) are equivalent:Corollary.LetKbe a tame AEC with a monster model. Assume thatKis stable in a proper class of cardinals. The following are equivalent:(1)For all high-enough λ,Khas no long splitting chains.(2)For all high-enough λ, there exists a good λ-frame on a skeleton ofKλ.(3)For all high-enough λ,Khas a unique limit model of cardinality λ.(4)For all high-enough λ,Khas a superlimit model of cardinality λ.(5)For all high-enough λ, the union of any increasing chain of λ-saturated models is λ-saturated.(6)There exists μ such that for all high-enough λ,Kis (λ,μ) -solvable.This gives evidence that there is a clear notion of superstability in the framework of tame AECs with a monster model. [ABSTRACT FROM AUTHOR]