The Hanf number for amalgamation of coloring classes

TL;DR

This paper investigates the amalgamation properties of coloring classes in abstract elementary classes, establishing their equivalence, determining the Hanf number for these properties, and improving existing results within ZFC.

Contribution

It proves the equivalence of amalgamation and disjoint amalgamation in coloring classes, finds the Hanf number for amalgamation, and enhances previous results by showing amalgamation holds up to  in ZFC.

Findings

Amalgamation and disjoint amalgamation are equivalent in coloring classes.

The Hanf number for amalgamation in coloring classes is established.

Amalgamation holds up to  in ZFC for classes studied by Baldwin, Kolesnikov, and Shelah.

Abstract

We study amalgamation properties in a family of abstract elementary classes that we call coloring classes. The family includes the examples previously studied in previous work of Baldwin, Kolesnikov, and Shelah. We establish that the amalgamation property is equivalent to the disjoint amalgamation property in all coloring classes; find the Hanf number for the amalgamation property for coloring classes; and improve the results of Baldwin, Kolesnikov, and Shelah by showing, in ZFC, that the (disjoint) amalgamation property for classes $K_\alpha$ studied in that paper must hold up to $\beth_\alpha$ (only a consistency result was previously known).