Independence, order, and the interaction of ultrafilters and theories

We consider the question, of longstanding interest, of realizing types in regular ultrapowers. In particular, this is a question about the interaction of ultrafilters and theories, which is both coarse and subtle. By our prior work it suffices to consider types given by instances of a single formula. In this article, we analyze a class of formulas φ whose associated characteristic sequence of hypergraphs can be seen as describing realization of first- and second-order types in ultrapowers on one hand, and properties of the corresponding ultrafilters on the other. These formulas act, via the characteristic sequence, as points of contact with the ultrafilter D, in the sense that they translate structural properties of ultrafilters into model-theoretically meaningful properties and vice versa. Such formulas characterize saturation for various key theories (e.g. Trg, Tfeq), yet their scope in Keisler’s order does not extend beyond Tfeq. The proof applies Shelah’s classification of second-order quantifiers.