Conant-independence and generalized free amalgamation

We initiate the study of a generalization of Kim-independence, Conant-independence, based on the notion of strong Kim-dividing of Kaplan, Ramsey and Shelah. A version of Conant-independence was originally introduced to prove that all [Formula: see text] theories are [Formula: see text]. 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 [Formula: see text] theories relative to a stationary independence relation. Generalizing Conant’s results on free amalgamation to the limits of our knowledge of the [Formula: see text] 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 [Formula: see text] holds is either [Formula: see text] or [Formula: see text] and is either simple or [Formula: see text]. We observe that these theories give an interesting class of examples of theories where Conant-independence is symmetric. This includes all of Conant’s examples, the small cycle-free random graphs of Shelah, and the (finite-language) [Formula: see text]-categorical Hrushovski constructions of Evans and Wong. We then answer Conant’s question on the existence of non-modular free amalgamation theories. We show that the generic functional structures of Kruckman and Ramsey are examples of non-modular free amalgamation theories. We also show that any free amalgamation theory is [Formula: see text] or [Formula: see text], while an [Formula: see text] free amalgamation theory is simple if and only if it is modular. Finally, we show that every theory where Conant-independence is symmetric is [Formula: see text]. Therefore, symmetry for Conant-independence gives the next known neostability-theoretic dividing line on the [Formula: see text] hierarchy beyond [Formula: see text]. We explain the connection to some established open questions.