Characterization of NIP theories by ordered graph-indiscernibles

We generalize the Unstable Formula Theorem characterization of stable theories from Shelah (1978) [11], , 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 [11], in our notation, a sequence of parameters from an L-structure M, (bi:i∈I), indexed by an L′-structure I is L′-generalized indiscernible in M if implies tp for all same-length, finite from I. Let Tg be the theory of linearly ordered graphs (symmetric, with no loops) in the language with signature Lg={<,R}. Let Kg be the class of all finite models of Tg. We show that a theory T has NIP if and only if any Lg-generalized indiscernible in a model of T indexed by an Lg-structure with age equal to Kg is an indiscernible sequence.