Characterization of NIP theories by ordered graph-indiscernibles

TL;DR

This paper extends the characterization of stable theories using indiscernible sequences to NIP theories via generalized indiscernibles indexed by ordered graph structures, providing a new criterion for NIP.

Contribution

It introduces a generalized form of indiscernibles indexed by ordered graphs and characterizes NIP theories through their behavior in this framework.

Findings

NIP theories characterized by ordered graph-indiscernibles

Generalized indiscernibles extend classical stability criteria

New criterion for NIP involving graph-structured indiscernibles

Abstract

We generalize the Unstable Formula Theorem characterization of stable theories from \citep{sh78}: that a theory $T$ is stable just in case any infinite indiscernible sequence in a model of $T$ is an indiscernible set. We use a generalized form of indiscernibles from \citep{sh78}: in our notation, a sequence of parameters from an $L$-structure $M$, $(b_i : i \in I)$, indexed by an $L'$-structure $I$ is \emph{$L'$-generalized indiscernible in $M$} if qftp$^{L'}(\ov{i};I)$=qftp$^{L'}(\ov{j};I)$ implies tp$^L(\ov{b}_{\ov{i}}; M)$ = tp$^L(\ov{b}_{\ov{j}};M)$ for all same-length, finite $\ov{i}, \ov{j}$ from $I$. Let $T_g$ be the theory of linearly ordered graphs (symmetric, with no loops) in the language with signature $L_g=\{<, R\}$. Let $\K_g$ be the class of all finite models of $T_g$. We show that a theory $T$ has NIP if and only if any $L_g$-generalized indiscernible in a model of $T$ indexed by an $L_g$-structure with age equal to $\K_g$ is an indiscernible sequence.