Model-theoretic dividing lines via posets

Abstract

We show that for each property $\mathsf{P}\in \{\mathsf{OP},\mathsf{IP},{\mathsf{TP}}_1,{\mathsf{TP}}_2,\mathsf{ATP},{\mathsf{SOP}}_3\}$ there is a poset ${\Sigma }_{\mathsf{P}}$ such that a theory has property $\mathsf{P}$ if and only if some model interprets a poset in which ${\Sigma }_{\mathsf{P}}$ can be embedded. We also introduce a new property $\mathsf{SUP}$, consistent with ${\mathsf{NIP}}_2$ and implying $\mathsf{ATP}$ and $\mathsf{SOP}$.