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.
• Version 2014-12-24_11 (60p) published version (61p)

@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},
 mrnumber = {3402699},
 mrclass = {03C20 (03C45 03E05 03E17)},
 doi = {10.1090/jams830},
 note = {\href{https://arxiv.org/abs/1208.5424}{arXiv: 1208.5424}},
 arxiv_number = {1208.5424}
}