Well-colorings and the Hanf number for amalgamation
The amalgamation property is a topic of fundamental interest in model theory and is still imperfectly understood. In the 1980s, Grossberg asked a question, which remains open to this day, about the existence of a Hanf number for amalgamation in abstract elementary classes. We introduce a new class of structures, called well-colorings, and use them to give a partial answer to Grossberg’s question, significantly improving upon previous work of Baldwin, Kolesnikov, and Shelah. We shall start the talk by briefly discussing the relevant model-theoretic definitions (no prior model-theoretic knowledge will be assumed) and will then give proofs of the main results, which are entirely set-theoretic and combinatorial in nature and of interest in their own right. This is joint work with Alexei Kolesnikov.