Recently Malliaris and Shelah (see their preprint http://math.uchicago.edu/~mem/Malliaris-Shelah-CST.pdf) have shown that theories with $SOP_2$ are maximal in the Keisler order. A preceding result of Shelah shows that theories with $SOP$ are maximal in this order: specifically Shelah in his book on classification theories gives a proof that the theory of $(\mathbb{Q},<)$ is maximal in Keisler order on theories. Is there another reference source where to look at this proof?
Recall that $T_1\leq T_2$ in Keisler order if whenever $G$ is a regular ultrafilter on $\lambda$, $M_i$ is model of $T_i$, and $M_2$ is such that $Ult(M_2,G)$ is $\lambda^+$-saturated, then also $Ult(M_1,G)$ is $\lambda^+$-saturated. Keisler showed that this relation defines indeed an order on theories (i.e. it is transitive and reflexive) and gave a set theoretic characterization of the maximal theories in terms of the combinatorial properties of the ultrafilters which saturate them (which are the good ultrafilters). There is still much work to do to understand the model theoretic properties of this order and Malliaris and Shelah's result is one of the recent most significant advances in this area. However it is not so easy to find in the literature a variety of results regarding the properties of this order. Any bibliographic indication on these matters is very welcome.
