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Characterization Theorems by Generalized Indiscernibles


We generalize the Unstable Formula Theorem characterization of stable theories in \cite{sh}: 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 \cite{sh,shnew}: in our notation, a sequence of parameters from an $L$-structure $M$, $(\ov{a}_i : i \in I)$, 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$^L(\ov{a}_{\ov{i}}; M)$ = tp$^L(\ov{a}_{\ov{j}}; M)$ for all same-length, finite $\ov{i}, \ov{j}$ from $I$. Let $T_g$ be the theory of linearly ordered symmetric graphs with no loops in the language with signature $\{<, R\}$, $L_g$. Say that a \emph{quantifier-free weakly-saturated} model of an $L$-theory $T$ is some model $M \vDash T$ that embeds realizations of all quantifier-free $L$-types consistent with $T$. We show that a theory $T$ is NIP if and only if every quantifier-free weakly-saturated $L_g$-generalized indiscernible in a model of $T$ is an indiscernible sequence.

In the process of proving this characterization, a result is introduced relating the utility of generalized indiscernibles indexed by a structure $I$ to the Ramsey-type properties of the structures in age($I$). Results about the previous development of generalized indiscernibles and known results about certain Ramsey-type properties are explicated in the text.

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