Building models in small cardinals in local abstract elementary classes

Abstract: 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.