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- W2901187712 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 $k < k'$, $T_{k+1, k} not trianglelefteq T_{k'+1, k'}$." @default.
- W2901187712 created "2018-11-29" @default.
- W2901187712 creator A5083331079 @default.
- W2901187712 date "2018-11-24" @default.
- W2901187712 modified "2023-09-27" @default.
- W2901187712 title "Amalgamation and Keisler's Order" @default.
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