# 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 However it is somewhat archaic and outdated (e.g. instead of writing $\prod_D \mathfrak{A}$, we writes $prod$-$D$ $\mathfrak{A}$).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) Chang 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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(4) is still wildly open; for (2) and (3), see theorem H in chapter 1 of Shelah's classification theory. I'm not sure what you mean by (1); if you meant "well-ordered" instead of "well defined," then that's also wildly open. EDIT: it seems immediate that the Keisler order is a partial order; am I missing something?

In general, Malliaris' thesis contains I think the best exposition and surveyn of the Keisler order, and page 3 of the thesis states Shelah's theorem H mentioned above.

I believe it is also wildly open whether the Keisler order is infinite, but I'm not sure.

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Sorry, I edited my post. I had remembered what I had read incorrectly. – Kyle Gannon Jul 23 '14 at 22:48
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
Furthermore, (4) is no longer open (as I have changed it). This is a result in Shelah's Classification theory (or which Malliaris attributes to Shelah's Classification theory). The point of this post is to find a single source which has all of these results together. – Kyle Gannon Jul 23 '14 at 22:57
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
Right, that is the correct definition of the Keisler order. However, one needs to show that this actually works, i.e. one needs to show that the ultrapowers are not model dependent, but theory dependent (since ultraproducts, are by defintion, constructions on models). This is done in Keisler's paper. – Kyle Gannon Jul 23 '14 at 23:22

In reply to the last comment by Noah, I may mention the following result by Malliaris-Shelah, from their paper KEISLER’S ORDER HAS INFINITELY MANY CLASSES ''

We prove, in ZFC, that there is an infinite strictly descending chain of classes of theories in Keisler’s order. Thus Keisler’s order is infinite and not a well order. Moreover, this chain occurs within the simple unstable theories, considered model-theoretically tame.

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