Characterization of NIP theories by ordered graph-indiscernibles

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 , ( b i : i ∈ I ) , indexed by an L ′ -structure I is L ′ -generalized indiscernible in M if qftp L ′ ( i ¯ ; I ) = qftp L ′ ( j ¯ ; I ) implies tp L ( b ¯ i ¯ ; M ) = tp L ( b ¯ j ¯ ; 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.