In this talk, I will define what it means for a coloring of substructures of an ultraproduct structure to be ``internal’’, and a definition 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.

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