Abstract
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 inM if qftpL′(i¯;I)=qftpL′(j¯;I) implies tpL(b¯i¯;M)=tpL(b¯j¯;M) for all same-length, finite i¯,j¯ 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.
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