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 However it is somewhat archaic and outdated (e.g. instead of writing $\prod\_D \mathfrak{A}$, we writes $prod$-$D$ $\mathfrak{A}$).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



References for the Keisler Order
(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.


