# References for the Keisler Order

Are there any good/modern references on the Keisler order. I have been reading Keisler's original paper, "Ultraproducts which are not Saturated", which introduces the order. However it is somewhat archaic and outdated (e.g. instead of writing $\prod_D \mathfrak{A}$, we writes $prod$-$D$ $\mathfrak{A}$). The paper also has some subtle errors and oddly cumbersome notation.

Furthermore, there are also sporadic "open-questions" spread throughout the paper and I believe many have been resolved since 1967. I gather this from tangential knowledge of the recent work on the order by Malliaris and Shelah.

$\mathbf{Searched}$: The following resources do not touch upon the Keisler order:

1) Change and Keisler's "Model Theory"

2) Marker's "Model Theory: An Introduction"

3) Hodges's "Model Theory"

4) Google searching any variant of "Keisler order Model Theory"

$\mathbf{Particulars}$: I am looking for a reference which does the following:

1) Proves the Keisler order is a partial order

2) Proves the existence of a maximal class

3) Proves the existence of a non-minimal class

4) Proves the order is linear for stable theories

First note that Keisler's original paper does not show that the order is linear, he lists this as an open question. I would be pretty surprised if there is not one collected work that does not do the four things listed above. However, I haven't been able to find one that does so.

Thanks

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Keisler's original paper is based on very much on models instead of theories. A lot of the results begin "Assume that $\mathfrak{A}$ is $\alpha^+$ saturated and $\mathfrak{B}$ is $\alpha^+$ universal". He does show, eventually, that the Keisler order is a partial order, but it is not immediately obvious why ultrapowers of two elementary equivalent models become saturated by exactly the same the same ultrafilters. –  Kyle Gannon Jul 23 '14 at 22:54
So the definition of Keisler order I'm familiar with is that $T_1\le T_2$ if whenever $U$ is a regular ultrafilter on $\lambda$, $M_i\models T_i$, and $Ult(M_2, U)$ is $\lambda^+$-saturated, then $Ult(M_1, U)$ is $\lambda^+$-saturated. Using this definition, poset-ness is immediate. I'm surprised to hear that (4) is no longer open; where in Shelah's book is that, or do you have another citation? Last I asked Maryanthe, she said (if I recall correctly) that it was still open. –  Noah Schweber Jul 23 '14 at 23:03