Some simple theories from a Boolean algebra point of view

We find a strong separation between two natural families of simple rank one theories in Keisler's order: the theories $T_m$ reflecting graph sequences, which witness that Keisler's order has the maximum number of classes, and the theories $T_{n,k}$, which are the higher-order analogues of the triangle-free random graph. The proof involves building Boolean algebras and ultrafilters “by hand” to satisfy certain model theoretically meaningful chain conditions. This may be seen as advancing a line of work going back through Kunen's construction of good ultrafilters in ZFC using families of independent functions. We conclude with a theorem on flexible ultrafilters, and open questions.