### Cofinality spectrum theorems in model theory, set theory and general topology

by Malliaris and Shelah. [MiSh:998]

J American Math Soc, 2016

We solve a set-theoretic problem (p=t) and show it has
consequences for the classification of unstable theories. That
is, we consider what pairs of cardinals (kappa_1, kappa_2)
may appear as the cofinalities of a cut in a regular ultrapower
of linear order, under the assumption that all symmetric
pre-cuts of cofinality no more than the size of the index set
are realized. We prove that the only possibility is (kappa, kappa^+)
where kappa is regular and kappa^+ is the
cardinality of the index set I . This shows that unless |I|
is the successor of a regular cardinal, any such ultrafilter must
be |I|^+-good. We then connect this work to the problem of
determining the boundary of the maximum class in Keisler's order.
Currently, SOP_3 is known to imply maximality. Here, we show
that the property of realizing all symmetric pre-cuts characterizes
the existence of paths through trees and thus realization of
types with SOP_2 (it was known that realizing {all} pre-cuts
characterizes realization of types with SOP_3). Thus whenever
lambda is not the successor of a regular cardinal, SOP_2
is lambda-maximal in Keisler's order. Moreover, the question
of the full maximality of SOP_2 is reduced to either
constructing a regular ultrafilter admitting the single asymmetric
cut described, or showing one cannot exist.

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