The Hanf number for amalgamation of coloring classes
  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). 
Alexei Kolesnikov and Chris Lambie-Hanson
math.LO