Saturating the random graph with an independent family of small range

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 $P(I)/\de$ 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 $M^I/\de_1$ is $\lambda^+$-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.