# Sh:1030

- Malliaris, M., & Shelah, S. (2016).
*Existence of optimal ultrafilters and the fundamental complexity of simple theories*. Adv. Math.,**290**, 614–681. arXiv: 1404.2919 DOI: 10.1016/j.aim.2015.12.009 MR: 3451934 -
Abstract:

We characterize the simple theories in terms of saturation of ultrapowers. This gives a dividing line at simplicity in Keisler’s order and gives a true outside definition of simple theories. Specifically, for any \lambda \geq \mu \geq \theta \geq \sigma such that \lambda = \mu^+, \mu = \mu^{<\theta} and \sigma is uncountable and compact (natural assumptions given our prior work, which allow us to work directly with models), we define a family of regular ultrafilters on \lambda called*optimal*, prove that such ultrafilters exist and prove that for any \mathcal{D} in this family and any M with countable signature, M^\lambda/\mathcal{D} is \lambda^+-saturated if Th(M) is simple and M^\lambda/ \mathcal{D} is not \lambda^+-saturated if Th(M) is not simple. The proof lays the groundwork for a stratification of simple theories according to the inherent complexity of coloring, and gives rise to a new division of the simple theories: (\lambda, \mu, \theta)-explicitly simple. - published version (68p)

Bib entry

@article{Sh:1030, author = {Malliaris, Maryanthe and Shelah, Saharon}, title = {{Existence of optimal ultrafilters and the fundamental complexity of simple theories}}, journal = {Adv. Math.}, fjournal = {Advances in Mathematics}, volume = {290}, year = {2016}, pages = {614--681}, issn = {0001-8708}, doi = {10.1016/j.aim.2015.12.009}, mrclass = {03C45 (03C20 03E05 03G05 06E10)}, mrnumber = {3451934}, mrreviewer = {Paolo Lipparini}, doi = {10.1016/j.aim.2015.12.009}, note = {\href{https://arxiv.org/abs/1404.2919}{arXiv: 1404.2919}}, arxiv_number = {1404.2919} }