An NIP-like Notion in Abstract Elementary Classes

Abstract

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 $\hat{I}»\ge LS(K)$, $K_{\hat{I}}»$ 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.