Some simple theories from a Boolean algebra point of view

TL;DR

This paper distinguishes two natural families of simple rank one theories in Keisler's order, using Boolean algebra constructions and ultrafilters, advancing understanding of model-theoretic classification.

Contribution

It introduces a new separation between theories reflecting graph sequences and higher-order triangle-free graphs in Keisler's order, using explicit Boolean algebra and ultrafilter constructions.

Findings

Separation between theories $T_rak{m}$ and $T_{n,k}$ in Keisler's order.

Construction of Boolean algebras satisfying specific model-theoretic chain conditions.

Development of ultrafilters with particular properties, including flexible ultrafilters.

Abstract

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.