Amalgamation and Keisler's Order

Malliaris and Shelah famously proved that Keisler’s order $⊴$ has infinitely many classes. In more detail, for each $2\le k<n<\omega$, let $T_{n,k}$ be the theory of the random $k$-ary $n$-clique free hypergraph. Malliaris and Shelah show that whenever $k+1<k^{\mathrm{\prime }}$, then $T_{k+1, k} \not \trianglelefteq T_{k’+1, k’}$\mathrm{.}However,theirargumentsdonotseparate$T_{k+1, k}$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 $3\le k<{\mathrm{\aleph }}_0$, 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 $k<k^{\mathrm{\prime }}$, $T_{k+1,k}⋬T_{k^{\mathrm{\prime }}+1,k^{\mathrm{\prime }}}$.