Model-theoretic properties of ultrafilters built by independent families of functions

Abstract

Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, {thus} saturating any stable theory. We then prove directly that a "bottleneck" in the inductive construction of a regular ultrafilter on $\lambda$ (i.e. a point after which all antichains of $P(\lambda)/D$ have cardinality less than $\lambda$) essentially prevents any subsequent ultrafilter from being flexible, {thus} from saturating any non-low theory. The paper's three main constructions are as follows. First, we construct a regular filter $D$ on $\lambda$ so that any ultrafilter extending $D$ fails to $\lambda^+$-saturate ultrapowers of the random graph, {thus} of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a measurable cardinal $\kappa$, we construct a regular ultrafilter on $\lambda > \kappa$ which is $\lambda$-flexible but not $\kappa^{++}$-good, improving our previous answer to a question raised in Dow 1975. Third, assuming a weakly compact cardinal $\kappa$, we construct an ultrafilter to show that $\lcf(\aleph_0)$ may be small while all symmetric cuts of cofinality $\kappa$ are realized. Thus certain families of pre-cuts may be realized while still failing to saturate any unstable theory.