Submission date: 14 October 2022

Abstract:

We initiate the study of a generalization of Kim-independence, Conant-independence, a version of which was introduced to prove that all NSOP_2 theories are NSOP_1. 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 NSOP_1 theories relative to a stationary independence relation. Generalizing Conant's results on free amalgamation to the limits of our knowledge of the 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 NSOP_1 holds is either NSOP_1 or NSOP_3 and is either simple or NTP_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) ω-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 NSOP_1 or SOP_3, while an 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 NSOP_4. Therefore, symmetry for Conant-independence gives the next known neostability-theoretic dividing line on the NSOP_n hierarchy beyond NSOP_1. We explain the connection to some established open questions.

Mathematics Subject Classification: 03C45

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Full text arXiv 2210.07527: pdf, ps.