There are many results in the literature where superstablity-like independence notions, without any categoricity assumptions, have been used to show the existence of larger models. In this paper we show that \emph{ stability} is enough to construct larger models for small cardinals assuming a mild locality condition for Galois types. $\mathbf{ Theorem.}$ Suppose $\lambda<2^{ \aleph_0}$. Let $\mathbf{ K}$ be an abstract elementary class with $\lambda \geq LS(\mathbf{ K})$. Assume $\mathbf{ K}$ has amalgamation in $\lambda$, no maximal model in $\lambda$, and is stable in $\lambda$. If $\mathbf{ K}$ is $(<\lambda^+, \lambda)$-local, then $\mathbf{ K}$ has a model of cardinality $\lambda^{ ++}$. The set theoretic assumption that $\lambda<2^{ \aleph_0}$ and model theoretic assumption of stability in $\lambda$ can be weakened to the model theoretic assumptions that $|\mathbf{ S}^{ na}(M)|< 2^{ \aleph_0}$ for every $M \in \mathbf{ K}_\lambda$ and stability for $\lambda$-algebraic types in $\lambda$. This is a significant improvement of Theorem 0.1., as the result holds on some unstable abstract elementary classes.
Categories: math.LO