Some applications of the real strict order property hierarchy

Abstract

We give applications of the properties $\mathrm{NSOP}_{r}$ for non-integer values of $r$ to problems on the original hierarchy $\mathrm{NSOP}_{n}$ for integer values of $n$. We first show that the properties $\mathrm{NSOP}_{r}$, previously defined for real values $r \geq 3$, are even well-defined for real values $r \geq 2$, showing that $\mathrm{NSOP}_{2} \subseteq \mathrm{NSOP}_{r}$ for our original definition of $\mathrm{NSOP}_{r}$ even when $2 < r < 3$. As a consequence, newness of all of the well-defined properties $\mathrm{NSOP}_{r}$ for non-integer $r$ would negatively resolve the problem of whether $\mathrm{NSOP}_{2}$ is equal to $\mathrm{NSOP}_{3}$. We then prove an approximate alternative between two possibilities: (1) that in extending Shelah's original $\mathrm{NSOP}_{n}$ hierarchy for integers $n \geq 3$ to the $\mathrm{NSOP}_{r}$ hierarchy for reals $r > 2$, we really did introduce new classification-theoretic properties, and (2) that $\mathrm{NSOP}_{n+1} \cap \mathrm{NTP}_{2} = \mathrm{NSOP}_{n} \cap \mathrm{NTP}_{2}$ for integers $n \geq 3$, which would resolve a central open problem in classification theory. More precisely, we give a rigorous sense in which (1) can fail on particularly general grounds, and then show that if (1) fails for these general reasons, (2) must be true. Finally, we apply cycle-removal techniques from the theory of the properties $\mathrm{NSOP}_{r}$ for real-values of $r$ to make progress on the question of whether $\mathrm{NSOP}_{2}$ is equal to $\mathrm{NSOP}_{3}$. We (a) show that if $\mathcal{H}$ is a hereditary class of structures defined by finitely many forbidden weakly embedded substructures, if every theory whose models have age $\mathcal{H}$ has $\mathrm{SOP}_{2}$, then every theory whose models have age $\mathcal{H}$ has $\mathrm{SOP}_{3}$, and (b) observe that we cannot replace $\mathrm{SOP}_{2}$ with $\mathrm{TP}$ here.