  We find a strong separation between two natural families of simple rank one
theories in Keisler's order: the theories $T_\mathfrak{m}$ 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.
