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An NIP-like Notion in Abstract Elementary Classes

Wentao Yang

TL;DR

The paper develops an analogue of NIP for abstract elementary classes (AECs) under set-theoretic assumptions, defining $K_\lambda$ as NIP when $|gS(M)|\le\mathrm{ded}\,\lambda$ for all $M\in K_\lambda$ and proving this is equivalent to the existence of a $w^*$-good $\lambda$-frame on $K$ (up to (Continuity)). When NIP fails, the authors show the theory can encode subsets, and they establish a Hanf-number-type phenomenon via Galois Morleyization and trees, indicating substantial combinatorial complexity in the unstable regime. The results leverage categoricity, tameness, and frame-extension techniques to connect a stability-like independence notion with a syntactic/semantic encoding mechanism, advancing neo-stability for non-elementary classes and providing a framework for extending nonforking frames to higher cardinals. Overall, the work clarifies when a robust forking-like theory exists in AECs and how failure of NIP manifests as powerful subset-encoding capabilities with implications for model-theoretic complexity.

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 $λ\geq LS(K)$, $K_λ$ 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.

An NIP-like Notion in Abstract Elementary Classes

TL;DR

The paper develops an analogue of NIP for abstract elementary classes (AECs) under set-theoretic assumptions, defining as NIP when for all and proving this is equivalent to the existence of a -good -frame on (up to (Continuity)). When NIP fails, the authors show the theory can encode subsets, and they establish a Hanf-number-type phenomenon via Galois Morleyization and trees, indicating substantial combinatorial complexity in the unstable regime. The results leverage categoricity, tameness, and frame-extension techniques to connect a stability-like independence notion with a syntactic/semantic encoding mechanism, advancing neo-stability for non-elementary classes and providing a framework for extending nonforking frames to higher cardinals. Overall, the work clarifies when a robust forking-like theory exists in AECs and how failure of NIP manifests as powerful subset-encoding capabilities with implications for model-theoretic complexity.

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 and , 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.
Paper Structure (4 sections, 11 theorems, 15 equations)

This paper contains 4 sections, 11 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The w*-good frame
  4. Syntactic independence property

Key Result

Theorem 1.2

Let $K$ be an AEC categorical in $\lambda\geq LS(K)$ , and $1\leq I(\lambda^+,K)<2^{\lambda^+}$. $K_\lambda$ has NIP if and only if there is a w*-good $\lambda$-frame on $K$ except possibly without (Continuity). Moreover,

Theorems & Definitions (56)

  • Definition 1.1
  • Theorem 1.2: $2^{\lambda^+}>2^\lambda$
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 46 more