Conant-independence and generalized free amalgamation

Abstract

We initiate the study of a generalization of Kim-independence, Conant-independence, based on the "strong Kim-dividing" of Kaplan, Ramsey and Shelah. We introduce an axiom on stationary independence relations essentially generalizing the "freedom" axiom in some of the free amalgamation theories of Conant, and show that this axiom provides the correct setting for carrying out arguments of Chernikov, Kaplan and Ramsey on $\mathrm{NSOP}_{1}$ theories relative to a stationary independence relation. Generalizing Conant's results on free amalgamation to the limits of our knowledge of the $\mathrm{NSOP}_{n}$ hierarchy, we show using methods from Conant as well as our previous work that any theory where the equivalent conditions of this local variant of $\mathrm{NSOP}_{1}$ holds is either $\mathrm{NSOP}_{1}$ or $\mathrm{SOP}_{3}$ and is either simple or $\mathrm{TP}_{2}$, and observe that these theories give an interesting class of examples of theories where Conant-independence is symmetric, including all of Conant's examples, the small cycle-free random graphs of Shelah and the (finite-language) $\omega$-categorical Hrushovski constructions of Evans and Wong. We then answer a question of Conant, showing that the generic functional structures of Conant and Kruckman are examples of non-modular free amalgamation theories, and show that any free amalgamation theory is $\mathrm{NSOP}_{1}$ or $\mathrm{SOP}_{3}$, while an $\mathrm{NSOP}_{1}$ free amalgamation theory is simple if and only if it is modular. Finally, we show that every theory where Conant-independence is symmetric is $\mathrm{NSOP}_{4}$. Therefore, symmetry for Conant-independence gives the next known neostability-theoretic dividing line on the $\mathrm{NSOP}_{n}$ hierarchy beyond $\mathrm{NSOP}_{1}$. We explain the connection to some established open questions.