# Sh:998

- Malliaris, M., & Shelah, S. (2016).
*Cofinality spectrum theorems in model theory, set theory, and general topology*. J. Amer. Math. Soc.,**29**(1), 237–297. arXiv: 1208.5424 DOI: 10.1090/jams830 MR: 3402699 -
Abstract:

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. - published version (61p)

Bib entry

@article{Sh:998, author = {Malliaris, Maryanthe and Shelah, Saharon}, title = {{Cofinality spectrum theorems in model theory, set theory, and general topology}}, journal = {J. Amer. Math. Soc.}, fjournal = {Journal of the American Mathematical Society}, volume = {29}, number = {1}, year = {2016}, pages = {237--297}, issn = {0894-0347}, doi = {10.1090/jams830}, mrclass = {03C20 (03C45 03E05 03E17)}, mrnumber = {3402699}, mrreviewer = {Klaas Pieter Hart}, doi = {10.1090/jams830}, note = {\href{https://arxiv.org/abs/1208.5424}{arXiv: 1208.5424}}, arxiv_number = {1208.5424} }