# The Hanf number for amalgamation

## Alexei Kolesnikov

### Towson University

In a joint work with Chris Lambie-Hanson, we study a family of abstract elementary classes (AEC) that we call coloring classes. Each coloring class is an AEC in a relational language $L$ containing exactly the $L$-structures whose finite substructures are isomorphic to one of the “allowed” finite structures. The work takes advantage of the fact that model-theoretic properties (e.g., existence of models and amalgamation) can be rephrased as properties of certain coloring functions. This allows us to improve the results of Baldwin, Kolesnikov, and Shelah: we show in ZFC that disjoint amalgamation can hold up to beth_{alpha}, alpha less than omega_1 (previously, only consistency results were known). We also give a partial answer to the question of Grossberg about the Hanf number for amalgamation property (not just disjoint amalgamation).