Resumen de Higher arity stability and the functional order property
A. Abd Aldaim, G. Conant, C. Terry
The k-dimensional functional order property (FOPk ) is a combinatorial property of a (k + 1)-partitioned formula. This notion arose in work of Terry and Wolf [59, 60], which identified NFOP2 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 NFOPk 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 NFOPk , 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 NFOPk is closed under Boolean combinations, and that FOPk 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 NFOP2 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 NFOP2 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.
