$n$-variable theorems for dividing lines characterized by positive consistency-inconsistency configurations, and their applications to preservation problems

Abstract

We define a class of classes of complete first-order theories, denoted by $\mathfrak{D}_{n\text{-var}}$, using positive consistency-inconsistency configurations and generalized indiscernibles. It contains the classes of stable, simple, NIP, NTP$_1$, NTP$_2$, NATP, NCTP, and NBTP theories, as well as the classes of theories not having a $(k,1,1)$-weave of depth $ω$ for any $k<ω$, theories not having an infinite $k$-grid for any $k<ω$, and NPM$^{(k)}$ theories for each $1<k<ω$. For any dividing line $D\in\mathfrak{D}_{n\text{-var}}$ and any complete first-order theory $T\notin D$, we can always find a formula $φ(x,y)$ witnessing $T\notin D$ with an indexed set of parameters such that the set of its instances required to be consistent has a realization whose algebraic dimension over the whole set of parameters is $|x|$. We call statements of this form $n$-variable theorems, as they may be regarded as weak versions of one-variable theorems. Under the appropriate hypotheses on $T$, we prove that $T\in D$ if and only if the corresponding theory belongs to $D$ for (i) $T^{gt}$, (ii) $T_P$, (iii) $T^{ind}$, (iv) $T^G_K$, (v) ACF$_T$, and (vi) any completion of $T^δ_g$. These are, respectively, the theories of generic trivializations, lovely pair expansions, $H$-structure expansions, vector spaces with a dense-codense generic $K$-subspace, algebraically closed fields with a distinguished subfield, and generic derivations of algebraically bounded fields. The $n$-variable theorem is essential in proving (i)--(v). We also introduce a larger class $\mathfrak{D}^h_{n\text{-var}}\supseteq\mathfrak{D}_{n\text{-var}}$, capturing higher-arity dividing lines such as NOP$_k$, NFOP$_k$, and NIP$_k$ for $1<k<ω$. The $n$-variable theorem and preservation results (ii), (iii), (iv), and (vi) also hold for this larger class, while (i) holds assuming ${\rm acl}={\rm dcl}$.