Amalgamation and Keisler's Order

TL;DR

This paper refines the understanding of Keisler's order by improving ultrafilter constructions and establishing new bounds, demonstrating that certain theories are not comparable within the order, especially for low theories and hypergraph theories.

Contribution

It provides uniform ultrafilter constructions and sharper bounds, extending the separation results in Keisler's order for a broader class of theories.

Findings

Sharper bounds for Keisler's order classes.

Separation of hypergraph theories in Keisler's order.

Application to low theories and independent amalgamation.

Abstract

Malliaris and Shelah famously proved that Keisler's order $\trianglelefteq$ has infinitely many classes. In more detail, for each $2 \leq k < n < \omega$, let $T_{n, k}$ be the theory of the random $k$-ary $n$-clique free hypergraph. Malliaris and Shelah show that whenever $k+1 < k'$, then $T_{k+1, k} \not \trianglelefteq T_{k'+1, k'}$. However, their arguments do not separate $T_{k+1, k}$ from $T_{k+2, k+1}$, and the model-theoretic properties detected by their ultrafilters are difficult to evaluate in practice. We uniformize the relevant ultrafilter constructions and obtain sharper model-theoretic bounds. As a sample application, we prove the following: suppose $3 \leq k < \aleph_0$, and $T$ is a countable low theory. Suppose that every independent system $(M_s: s \subsetneq k)$ of countable models of $T$ can be independently amalgamated. Then $T_{k, k-1} \not \trianglelefteq T$. In particular, for all $k < k'$, $T_{k+1, k} \not \trianglelefteq T_{k'+1, k'}$.