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Preprint Number 500
500. M. Malliaris and S. Shelah
Saturating the random graph with an independent family of small range
Submission date: 28 August 2012.
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| = λ > \aleph_0, the fact that P(I)/D has little freedom (as measured by the fact that any maximal antichain is of size <λ, 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 λ^+-saturated whenever M is a model of the theory of the random graph. 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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