Saturating the random graph with an independent family of small range

by Malliaris and Shelah. [MiSh:1009]
Logic without borders, 2015
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 {D} on I, |I| = lambda > aleph_0, the fact that {{P}}(I)/ {D} has little freedom (as measured by the fact that any maximal antichain is of size < lambda, or even countable) does not prevent extending {D} to an ultrafilter {D}1 on I which saturates ultrapowers of the random graph. ``Saturates'' means that M^I/ {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.

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