We generalize the Unstable Formula Theorem characterization of stable theories from [1]: 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 [1]: in our notation, a sequence of parameters from an L-structure M,for all same-length, finite i, 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.
We give definitions that distinguish between two notions of indiscernibility
for a set $\{a_\eta \mid \eta \in \W\}$ that saw original use in \cite{sh90},
which we name \textit{$\s$-} and \textit{$\n$-indiscernibility}. Using these
definitions and detailed proofs, we prove $\s$- and $\n$-modeling theorems and
give applications of these theorems. In particular, we verify a step in the
argument that TP is equivalent to TP$_1$ or TP$_2$ that has not seen
explication in the literature. In the Appendix, we exposit the proofs of
\citep[{App. 2.6, 2.7}]{sh90}, expanding on the details.Comment: submitte
We use the notion of collapse of generalized indiscernible sequences to classify various model theoretic dividing lines. In particular, we use collapse of n-multi-order indiscernibles to characterize op-dimension n; collapse of function-space indiscernibles (i.e. parameterized equivalence relations) to characterize rosy theories; and finally, convex equivalence relation indiscernibles to characterize NTP2 theories.
Abstract. It was shown in [16] that for a certain class of structures I, Iindexed indiscernible sets have the modeling property just in case the age of I is a Ramsey class. We expand this known class of structures from ordered structures in a finite relational language to ordered, locally finite structures which isolate quantifier-free types by way of quantifier-free formulas. As a corollary, we obtain a new Ramsey class of finite trees.
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