A Study of Abstract Elementary Classes in the context of Graphs

In the framework of graphs, we study abstract elementary classes (aecs). In this work we analyze several properties of Forb(G) and versions of Forb-Con(G) in the context of aecs and we present some examples of classes of graphs which contradicts amalgamation property.Comment: 17 page

Building models in small cardinals in local abstract elementary classes

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 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 $\lambda \geq \operatorname{LS}(\mathbf{K})$. 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 weaken 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 $\lambda$. We further use the result just mentioned to provide a positive answer to Grossberg's question for small cardinals assuming a mild locality condition for Galois types and without any stability assumptions. This last result relies on an unproven claim of Shelah (Fact 4.5 of this paper) which we were unable to verify

An NIP-like Notion in Abstract Elementary Classes

This paper is a contribution to "neo-stability" type of result for abstract elementary classes. Under certain set theoretic assumptions, we propose a definition and a characterization of NIP in AECs. The class of AECs with NIP properly contains the class of stable AECs. We show that for an AEC $K$ and $\lambda\geq LS(K)$, $K_\lambda$ is NIP if and only if there is a notion of nonforking on it which we call a w*-good frame. On the other hand, the negation of NIP leads to being able to encode subsets