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I posted it on MSE and didn't receive any comments or answers, so I thought I would post it here.

(See https://math.stackexchange.com/questions/895549/keisler-order-saturated-ultrapowers)

Keisler's paper "Ultraproducts which are not Saturated" states the following theorem as a corollary to a (much more) generalized theorem. However, I cannot, for the life of me, figure out how to prove it for the specific case. Furthermore, Malliaris's Thesis cites Shelah's Classification Theory for the proof. I looked there and found a discussion of games (which I couldn't quite follow).

Theorem: Let $D$ be a regular ultrafilter over $I$ (where $|I| =\alpha$) and let $\mathfrak{A} \equiv \mathfrak{B}$. Then the ultrapowers $\prod_D\mathfrak{A}$ is $\alpha^+$-saturated iff $\prod_D \mathfrak{B}$ is $\alpha^+$-saturated.

Motivation: Proving this theorem shows that the Keisler Order is a well-defined order on theories (as opposed to an order on models).

Thanks

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W.W. Comfort, S. Negrepontis, The Theory of Ultrafilters, section 13, in particular Theorem 13.7 and Corollary 13.8, might be useful to you. It contains a textbook presentation of the relevant proofs (without games).

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