The k-dimensional functional order property ( \(\operatorname {FOP}_k\) ) is a combinatorial property of a \((k+1)\) -partitioned formula. This notion arose in work of Terry and Wolf [59, 60], which identified \(\operatorname {NFOP}_2\) 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 \(\operatorname {NFOP}_k\) 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 \(\operatorname {NIP}\) . Our results include several new characterizations of \(\operatorname {NFOP}_k\) , 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 \(\operatorname {NFOP}_k\) is closed under Boolean combinations, and that \(\operatorname {FOP}_k\) 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 \(\operatorname {NFOP}_2\) 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 \(\operatorname {NFOP}_2\) 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.
