On the properties $\mathrm{SOP}_{2^{n+1}+1}$

Abstract

We show that approximations of strict order can calibrate the fine structure of genericity. Particularly, we find exponential behavior within the $\mathrm{NSOP}_{n}$ hierarchy from model theory. Let $0$-$ð$-independence denote forking-independence. Inductively, a formula $(n+1)$-$ð$-divides over $M$ if it divides by every $n$-$ð$-independent Morley sequence over $M$, and $(n+1)$-$ð$-forks over $M$ if it implies a disjunction of formulas that $(n+1)$-$ð$-divide over $M$; the associated independence relation over models is called $(n+1)$-$ð$-independence. We show that a theory where $n$-$ð$-independence is symmetric or transitive must be $\mathrm{NSOP}_{2^{n+1}+1}$. We then show that, in the classical examples of $\mathrm{NSOP}_{2^{n+1}+1}$ theories, $n$-$ð$-independence is symmetric and transitive; in particular, there are strictly $\mathrm{NSOP}_{2^{n+1}+1}$ theories where $n$-$ð$-independence is symmetric and transitive, leaving open the question of whether symmetry or transitivity of $n$-$ð$-independence is equivalent to $\mathrm{NSOP}_{2^{n+1}+1}$.