Tameness, Uniqueness and amalgamation

We combine two approaches to the study of classification theory of AECs: 1. that of Shelah: studying non-forking frames without assuming the amalgamation property but assuming the existence of uniqueness triples and 2. that of Grossberg and VanDieren: (studying non-splitting) assuming the amalgamation property and tameness. In [JrSh875], we derive a good non-forking $λ^+$-frame from a semi-good non-forking $λ$-frame. But the classes $K_{λ^+}$ and $\preceq \restriction K_{λ^+}$ are replaced: $K_{λ^+}$ is restricted to the saturated models and the partial order $\preceq \restriction K_{λ^+}$ is restricted to the partial order $\preceq^{NF}_{λ^+}$. Here, we avoid the restriction of the partial order $\preceq \restriction K_{λ^+}$, assuming that every saturated model (in $λ^+$ over $λ$) is an amalgamation base and $(λ,λ^+)$-tameness for non-forking types over saturated models, (in addition to the hypotheses of [JrSh875]): We prove that $M \preceq M^+$ if and only if $M \preceq^{NF}_{λ^+}M^+$, provided that $M$ and $M^+$ are saturated models. We present sufficient conditions for three good non-forking $λ^+$-frames: one relates to all the models of cardinality $λ^+$ and the two others relate to the saturated models only. By an `unproven claim' of Shelah, if we can repeat this procedure $ω$ times, namely, `derive' good non-forking $λ^{+n}$ frame for each $n<ω$ then the categoricity conjecture holds. Vasey applies one of our main theorems in a proof of the categoricity conjecture under the above `unproven claim' of Shelah and more assumptions. In [Jrprime], we apply the main theorem in a proof of the existence of primeness triples.

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