### 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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