Amalgamation and Keisler's Order

Abstract

Malliaris and Shelah famously proved that Keisler's order $⊴$ has infinitely many classes. In more detail, for each $2\le k<n<\ddot{I}‰$, let $T_{n,k}$ be the theory of the random $k$-ary $n$-clique free hypergraph. Malliaris and Shelah show that whenever $k+1<k^{\prime }$, then $T_{k+1,k}/⊴T_{k^{\prime }+1,k^{\prime }}$. 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 $3\le k<{\aleph }_0$, and $T$ is a countable low theory. Suppose that every independent system $(M_s:s⊊k)$ of countable models of $T$ can be independently amalgamated. Then $T_{k,k-1}/⊴T$. In particular, for all $k<k^{\prime }$, $T_{k+1,k}/⊴T_{k^{\prime }+1,k^{\prime }}$.

43 pages