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 Dn−var, using positive consistency-inconsistency configurations and generalized indiscernibles. It contains the classes of stable, simple, NIP, NTP1, NTP2, NATP, NCTP, NBTP, NWP, NGP, and NPM(k) theories. For any dividing line D∈Dn−var and any complete first-order theory T∉ D, we can always find a formula φ(x,y) witnessing T∉ 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∈ D if and only if the corresponding theory belongs to D for (i) Tgt, (ii) TP, (iii) Tind, (iv) TGK, (v) ACFT, and (vi) any completion of Tδ̅,ncg. 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 non-commutative derivations of algebraically bounded fields. The n-variable theorem is essential in proving (i)--(v). We also prove preservation for a special case of Mordell-Lang theories. We extend the framework to a larger class Dhn−var⊇Dn−var, capturing higher-arity dividing lines such as NOPk, NFOPk, and NIPk 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 acl=dcl.
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BibTeX
@misc{kim2026nvariable,
title = {{$n$-variable theorems for dividing lines characterized by positive consistency-inconsistency configurations, and their applications to preservation problems}},
author = {Joonhee Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.36394},
eprint = {2609.36394},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36394},
}
RIS
TY - UNPB
TI - n-variable theorems for dividing lines characterized by positive consistency-inconsistency configurations, and their applications to preservation problems
AU - Joonhee Kim
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2609.36394
UR - https://arxiv.org/abs/2609.36394
AB - We define a class of classes of complete first-order theories, denoted by D_n−var, using positive consistency-inconsistency configurations and generalized indiscernibles. It contains the classes of stable, simple, NIP, NTP_1, NTP_2, NATP, NCTP, NBTP, NWP, NGP, and NPM^(k) theories. For any dividing line D∈D_n−var and any complete first-order theory T∉ D, we can always find a formula φ(x,y) witnessing T∉ 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∈ 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^δ̅,nc_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 non-commutative derivations of algebraically bounded fields. The n-variable theorem is essential in proving (i)--(v). We also prove preservation for a special case of Mordell-Lang theories. We extend the framework to a larger class D^h_n−var⊇D_n−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 acl=dcl.
ER -