# Sh:1009

• Malliaris, M., & Shelah, S. (2015). Saturating the random graph with an independent family of small range. In A. Hirvonen, M. Kesala, J. Kontinen, R. Kossak, & A. Villaveces, eds., Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics, Berlin, Boston: De Gruyter, pp. 319–337.
• Abstract:
Motivated by Keisler’s order, a far-reaching program of understanding basic model-theoretic structure through the lens of regular ultrapowers, we prove that for a class of regular filters \mathcal{D} on I, |I| = \lambda > \aleph_0, the fact that {\mathcal{P}}(I)/ \mathcal{D} has little freedom (as measured by the fact that any maximal antichain is of size <\lambda, or even countable) does not prevent extending \mathcal{D} to an ultrafilter \mathcal{D}1 on I which saturates ultrapowers of the random graph. “Saturates” means that M^I/\mathcal{D}_1 is \lambda^+-saturated whenever M \models T_{\mathbf{rg}}. This was known to be true for stable theories, and false for non-simple and non-low theories. This result and the techniques introduced in the proof have catalyzed the authors’ subsequent work on Keisler’s order for simple unstable theories. The introduction, which includes a part written for model theorists and a part written for set theorists, discusses our current program and related results.
• Version 2014-01-27_11 (15p) published version (20p)
Bib entry
@incollection{Sh:1009,
author = {Malliaris, Maryanthe and Shelah, Saharon},
title = {{Saturating the random graph with an independent family of small range}},
booktitle = {{Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic and Philosophy of Mathematics}},
series = {Ontos Mathematical Logic, vol. 5},
year = {2015},
pages = {319--337},
publisher = {Berlin, Boston: De Gruyter},
editor = {A. Hirvonen and M. Kesala and J. Kontinen and R. Kossak and A. Villaveces},
doi = {10.1515/9781614516873.319},
note = {\href{https://arxiv.org/abs/1208.5585}{arXiv: 1208.5585}},
arxiv_number = {1208.5585}
}