Non-forking w-good frames
We introduce the notion of a w-good $λ$-frame which is a weakening of Shelah's notion of a good $λ$-frame. Existence of a w-good $λ$-frame implies existence of a model of size $λ^{++}$. Tameness and amalgamation imply extension of a w-good $λ$-frame to larger models. As an application we show: $\textbf{Theorem}$ Suppose $2^λ< 2^{λ^{+}} < 2^{λ^{++}}$ and $2^{λ^{+}} > λ^{++}$. If $I(K, λ) = I(K, λ^{+}) = 1 \leq I(K, λ^{++}) < 2^{λ^{++}}$ and $K$ is $(λ, λ^+)$-tame, then $K_{λ^{+++}} \neq \emptyset$. The proof presented clarifies some of the details of the main theorem of [Sh576] and avoids using the heavy set-theoretic machinery of [Sh: h §VII] by replacing it with tameness.
math.LO↗