Model Theory Seminar

I will present a transfer principle in structural Ramsey theory from finite structures to ultraproducts. In joint work with Bartosova, Dzamonja and Scow, we show that under certain mild conditions and assuming CH, when a class of finite structures has finite small Ramsey degrees, the ultraproduct has finite big Ramsey degrees for internal colorings. All Ramsey-theoretic definitions will be provided, and if time permits, I will give a sketch of the proof.

In this talk, I plan to explore a few notions of 'genericity' in the context of models of arithmetic. I will recall the notion of genericity borrowed from set-theory, used by Simpson to prove that every countable model of PA has an expansion to a pointwise definable model of PA*. I will then explore other notions of genericity inspired by more model-theoretic contexts. One such notion is 'neutrality': in a model M, we say an undefinable set X is neutral if the definable closure relation in (M, X) is the same as in M. Another notion, inspired by work done on model-theoretic genericity by Chatzidakis and Pillay, is called CP-genericity. I will explore these notions and outline some results, including: (1) every model of PA has a neutral set which is not CP-generic, (2) every countable model of PA has a CP-generic which is not neutral (and in fact, fails neutrality spectacularly: ie, we can find a CP-generic where the expansion is pointwise definable), and (3) every countable model of PA has a neutral CP-generic. This talk touches on work contained in two papers, one of which was joint work with Roman Kossak, and the other was joint work with James Schmerl.