Characterization of NIP theories by ordered graph-indiscernibles

Annals of Pure and Applied Logic 163 (11):1624-1641 (2012)
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Abstract

We generalize the Unstable Formula Theorem characterization of stable theories from Shelah [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, , indexed by an L′-structure I is L′-generalized indiscernible inM if qftpL′=qftpL′ implies tpL=tpL for all same-length, finite ¯,j from I. Let Tg be the theory of linearly ordered graphs 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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References found in this work

On ◁∗-maximality.Mirna Džamonja & Saharon Shelah - 2004 - Annals of Pure and Applied Logic 125 (1-3):119-158.
Models without indiscernibles.Fred G. Abramson & Leo A. Harrington - 1978 - Journal of Symbolic Logic 43 (3):572-600.
The stability spectrum for classes of atomic models.John T. Baldwin & Saharon Shelah - 2012 - Journal of Mathematical Logic 12 (1):1250001-.
Karp complexity and classes with the independence property.M. C. Laskowski & S. Shelah - 2003 - Annals of Pure and Applied Logic 120 (1-3):263-283.

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