Onn-dependent groups and fields II

Abstract

We continue the study of n-dependent groups, fields and related structures, largely motivated by the conjecture that every n-dependent field is dependent. We provide evidence toward this conjecture by showing that every infinite n-dependent valued field of positive characteristic is henselian, obtaining a variant of Shelah’s Henselianity Conjecture in this case and generalizing a recent result of Johnson for dependent fields. Additionally, we prove a result on intersections of type-definable connected components over generic sets of parameters in n-dependent groups, generalizing Shelah’s absoluteness of $G^{00}$ in dependent theories and relative absoluteness of $G^{00}$ in $2$ -dependent theories. In an effort to clarify the scope of this conjecture, we provide new examples of strictly $2$ -dependent fields with additional structure, showing that Granger’s examples of non-degenerate bilinear forms over dependent fields are $2$ -dependent. Along the way, we obtain some purely model-theoretic results of independent interest: we show that n-dependence is witnessed by formulas with all but one variable singletons; provide a type-counting criterion for $2$ -dependence and use it to deduce $2$ -dependence for compositions of dependent relations with arbitrary binary functions (the Composition Lemma); and show that an expansion of a geometric theory T by a generic predicate is dependent if and only if it is n-dependent for some n, if and only if the algebraic closure in T is disintegrated. An appendix by Martin Bays provides an explicit isomorphism in the Kaplan-Scanlon-Wagner theorem.

Sets, groups, and fields definable in vector spaces with a bilinear form

“…In [3] it was proved that (the completions of) the theories of vector spaces with a nondegenerate bilinear form over an NIP (another tameness property studied extensively in model theory) field satisfy a generalisation of NIP called NIP 2 ; in particular, T ∞ and T RCF ∞ (see the paragraph below) are examples of NIP 2 theories which are not NIP.…”

An improved bound for regular decompositions of 3-uniform hypergraphs of bounded VC2-dimension

“…The notion of VC 2 -dimension was first introduced by Shelah [22], who also studied it in the context of groups [23]. It was later shown to have nice modeltheoretic characterizations by Chernikov, Palacin, and Takeuchi [5] to have further natural connections to groups and fields by Hempel and Chernikov [15;2], and to have applications in combinatorics by the author [25].…”

An improved bound for regular decompositions of $3$-uniform hypergraphs of bounded $VC_2$-dimension

“…The notion of VC 2 -dimension was first introduced [23] by Shelah, who also studied it in the context of groups [22]. It was later shown to have nice model theoretic characterizations by Chernikov-Palacin-Takeuchi [4], to have further natural connections to groups and fields by Hempel and Chernikov-Hempel [3,15], and to have applications in combinatorics by the author [27].…”