Weakening the local character

TL;DR

This paper enhances a non-forking relation in Abstract Elementary Classes to satisfy local character, aligning it more closely with stable first-order theories and impacting the understanding of model cardinalities.

Contribution

It improves Shelah's non-forking relation to include local character, bridging properties between AECs and superstable first-order theories.

Findings

Non-forking relation now satisfies local character.

The function counting models is constrained by this property.

Enhanced relation aligns AECs with stable first-order theory properties.

Abstract

In [Sh E46], Shelah obtained a non-forking relation for an AEC, (K,\preceq), with LST-number at most \lambda, which is categorical in \lambda and \lambda^+ and has less than 2^{\lambda^+} models of cardinality \lambda^{++}, but at least one. This non-forking relation satisfies the main properties of the non-forking relation on stable first order theories, but only a weak version of the local character. Here, we improve this non-forking relation such that it satisfies the local character, too. Therefore it satisfies the main properties of the non-forking relation on superstable first order theories. We conclude that the function \lambda \to I(\lambda,K), which assigns to each cardinal \lambda, the number of models in K of cardinality \lambda, is not arbitrary.