Lynn Scow (California State University, San Bernardino)

Big Ramsey degrees for internal colorings

In this talk, I will define what it means for a coloring of substructures of an ultraproduct structure to be "internal", and a notion of finite big Ramsey degree for internal colorings. I will also present a certain Ramsey degree transfer theorem from countable sequences of finite structures to their ultraproducts, assuming AC and some additional mild assumptions. The big Ramsey degree of a finite structure in an ultraproduct can differ markedly from its internal big Ramsey degree, as demonstrated by the example of the class of all finite linear orders, which I will explain.

This is joint work with Dana Bartošová, Mirna Džamonja and Rehana Patel.