Abstract independence relations in neostability theory

We develop a framework, in the style of Adler, for interpreting the notion of “witnessing” that has appeared (usually as a variant of Kim’s Lemma) in different areas of neostability theory as a binary relation between abstract independence relations. This involves extending the relativisations of Kim-independence and Conant-independence due to Mutchnik to arbitrary independence relations. After developing this framework, we show that several results from simplicity, $\text{NTP}_2$, $\text{NSOP}_1$, and beyond follow as instances of general theorems for abstract independence relations. In particular, we prove the equivalence between witnessing and symmetry and the implications from this notion to chain local character and the weak independence theorem, and recover some partial converses. Finally, we use this framework to prove a dichotomy between $\text{NSOP}_1$ and Kruckman and Ramsey’s $\text{BTP}$ that applies to most known $\text{NSOP}_4$ examples in the literature.

💡 Research Summary

This paper develops a unifying framework for the notion of “witnessing”—often presented as variants of Kim’s Lemma—in contemporary neostability theory. The author treats witnessing as a binary relation between abstract independence relations, extending the relativisations of Kim‑independence and Conant‑independence (originally introduced by Mutchnik) to arbitrary independence relations.

The work begins by recalling Adler’s abstract independence relations, a ternary relation ⊥ on small subsets of a monster model that satisfies a collection of axioms such as monotonicity, existence, extension, normality, transitivity, strong finite character, local character, symmetry, stationarity, and the independence theorem. The paper then introduces several operations on such relations: the “*” operation that forces strong finite character, the opposite operation ⊥ᵒᵖ that swaps the two outer arguments, and the left‑extension operation ⊥ˡᵉ. These operations preserve or strengthen the usual axioms and allow one to move between different concrete independence notions (forking, dividing, Kim‑forking, etc.) in a systematic way.

In Section 4 the author defines a relativised version of Kim‑independence for an arbitrary base independence relation ⊥. Roughly, ⊥‑Kim‑independence holds when every ⊥‑Morley sequence witnesses ⊥‑Kim‑dividing. An analogous relativisation of Conant‑independence is also given.

Section 5 introduces three historically distinct formulations of witnessing and proves that they are equivalent under mild hypotheses. The unified definition, called the Generalised Universal Witnessing Property (GUWP), says that ⊥₁‑Kim‑independence is witnessed by all ⊥₂‑Morley sequences. This captures the essence of Kim’s Lemma in a model‑theoretic language that is independent of the particular theory.

The paper then explores the consequences of GUWP. In Section 6 it is shown that if an independence relation ⊥ satisfies full existence, monotonicity, strong finite character, and right extension, then its Conant‑relativisation automatically has chain local character. Section 7 establishes the equivalence between GUWP and symmetry for ⊥‑Kim‑independence (Theorem 7.1). Consequently, symmetry—traditionally proved via intricate tree constructions—can be derived purely from the semantic properties of abstract independence relations.

Section 8 proves a general “weak independence theorem”. Assuming GUWP and a weak transitivity condition, one obtains the weak independence theorem for the relativised Kim‑independence (Theorem 8.7). Moreover, a partial converse (Theorem 8.16) shows that if ⊥₂‑Conant‑independence satisfies the weak independence theorem over models, then ⊥₁ enjoys GUWP with respect to ⊥₂. This yields a new, purely semantic proof that Conant‑independence satisfying the independence theorem forces the ambient theory to be NSOP₁, bypassing the classical tree‑construction argument of Chernikov–Ramsey.

Finally, Section 9 delivers a dichotomy between NSOP₁ and the Branching Tree Property (BTP). Under the hypotheses that an independence relation ⊥ has full existence, monotonicity, and either stationarity or quasi‑strong finite character, and that ⊥ has GUWP with respect to itself, the theory either is NSOP₁ or exhibits BTP. This result subsumes many known examples: most published NSOP₄ theories turn out to have BTP, confirming a pattern observed in the literature.

Overall, the paper shows that many disparate results across simplicity, NTP₂, NSOP₁, and higher NSOPₙ can be obtained as instances of a small collection of abstract theorems about independence relations and witnessing. By recasting witnessing as a relation between independence relations, the author provides a powerful, theory‑independent toolkit for future investigations of dividing lines, and opens the way to extending these results to arbitrary bases beyond models.