Submission date: 11 August 2021

Abstract:

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.

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Full text arXiv 2108.05314: pdf, ps.