Submission date: 25 June 2011.

Abstract:

We generalize the Unstable Formula Theorem characterization of stable theories from [1]: 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 [1]: in our notation, a sequence of parameters from an L-structure M, (b_i : i \in 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.

Mathematics Subject Classification: 03C45, 03C68, 05C55

Keywords and phrases: Classification theory, Ramsey classes, Generalized indiscernibles

Full text arXiv 1106.5153: pdf, ps.