Higher arity stability and the functional order property

Abstract

The $k$-dimensional functional order property (${\text{FOP}}_k$) is a combinatorial property of a $(k+1)$-partitioned formula. This notion arose in work of Terry and Wolf, which identified ${\text{NFOP}}_2$ as a ternary analogue of stability in the context of two finitary combinatorial problems related to hypergraph regularity and arithmetic regularity. In this paper we show ${\text{NFOP}}_k$ has equally strong implications in model-theoretic classification theory, where its behavior as a $(k+1)$-ary version of stability is in close analogy to the behavior of $k$-dependence as a $(k+1)$-ary version of $\text{NIP}$. Our results include several new characterizations of ${\text{NFOP}}_k$, including a characterization in terms of collapsing indiscernibles, combinatorial recharacterizations, and a characterization in terms of type-counting when $k=2$. As a corollary of our collapsing theorem, we show ${\text{NFOP}}_k$ is closed under Boolean combinations, and that ${\text{FOP}}_k$ can always be witnessed by a formula where all but one variable have length $1$. When $k=2$, we prove a composition lemma analogous to that of Chernikov and Hempel from the setting of $2$-dependence. Using this, we provide a new class of algebraic examples of ${\text{NFOP}}_2$ theories. Specifically, we show that if $T$ is the theory of an infinite dimensional vector space over a field $K$, equipped with a bilinear form satisfying certain properties, then $T$ is ${\text{NFOP}}_2$ if and only if $K$ is stable. Along the way we provide a corrected and reorganized proof of Granger's quantifier elimination and completeness results for these theories.

71 pages, several corrections and updates to reflect recent literature, final version following referee report