# Sh:1051

- Malliaris, M., & Shelah, S. (2017).
*Model-theoretic applications of cofinality spectrum problems*. Israel J. Math.,**220**(2), 947–1014. arXiv: 1503.08338 DOI: 10.1007/s11856-017-1526-7 MR: 3666452 -
Abstract:

We apply the recently developed technology of cofinality spectrum problems to prove a range of theorems in model theory. First, we prove that any model of Peano arithmetic is \lambda-saturated iff it has cofinality \geq \lambda and the underlying order has no (\kappa, \kappa)-gaps for regular \kappa < \lambda. We also answer a question about balanced pairs of models of PA. Second, assuming instances of GCH, we prove that SOP_2 characterizes maximality in the interpretability order \triangleleft^*, settling a prior conjecture and proving that SOP_2 is a real dividing line. Third, we establish the beginnings of a structure theory for NSOP_2, proving that NSOP_2 can be characterized by the existence of few so-called higher formulas. In the course of the paper, we show that \mathfrak{p}_{\mathbf{s}} = \mathfrak{t}_{\mathbf{s}} in*any*weak cofinality spectrum problem closed under exponentiation (naturally defined). We also prove that the local versions of these cardinals need not coincide, even in cofinality spectrum problems arising from Peano arithmetic. *2015.02.12 The new note from Feb on the delayed lemma make no sense:- the finite satisfiability ensure the formulas are compatible. Look back at the old.. The second theorem which is delayed (union of few pairwise non-contradictory) - a debt, look again - published version (68p)

Bib entry

@article{Sh:1051, author = {Malliaris, Maryanthe and Shelah, Saharon}, title = {{Model-theoretic applications of cofinality spectrum problems}}, journal = {Israel J. Math.}, fjournal = {Israel Journal of Mathematics}, volume = {220}, number = {2}, year = {2017}, pages = {947--1014}, issn = {0021-2172}, mrnumber = {3666452}, mrclass = {03E04 (03C20 03C45 03C55 03E05 03E17)}, doi = {10.1007/s11856-017-1526-7}, note = {\href{https://arxiv.org/abs/1503.08338}{arXiv: 1503.08338}}, arxiv_number = {1503.08338} }