Characterization of NIP theories by ordered graph-indiscernibles

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 $L^{\mathrm{\prime }}$-structure $I$ is \emph{$L^{\mathrm{\prime }}$-generalized indiscernible in $M$} if qftp${}^{L^{\mathrm{\prime }}}(\text{ov}i;I)$=qftp${}^{L^{\mathrm{\prime }}}(\text{ov}j;I)$ implies tp${}^L(\text{ov}{b}_{\text{ov}i};M)$ = tp${}^L(\text{ov}{b}_{\text{ov}j};M)$ for all same-length, finite $\text{ov}i,\text{ov}j$ 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 ${\text{K}}_g$ 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 ${\text{K}}_g$ is an indiscernible sequence.