Submission date: 22 May 2023

Abstract:

The k-dimensional functional order property (FOP_k) is a combinatorial property of a (k+1)-partitioned formula. This notion arose in work of Terry and Wolf, which identified 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 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 NIP. Our results include several new characterizations of 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 NFOP_k is closed under Boolean combinations, and that 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 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 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.

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Full text arXiv 2305.13111: pdf, ps.