Worst case expansions of complete theories

Abstract

Given a complete theory $T$ and a subset $Y\subseteq X^k$, we precisely determine the {\em worst case complexity}, with respect to further monadic expansions, of an expansion $(M,Y)$ by $Y$ of a model $M$ of $T$ with universe $X$. In particular, although by definition monadically stable/NIP theories are robust under arbitrary monadic expansions, we show that monadically NFCP (equivalently, mutually algebraic) theories are the largest class that is robust under anything beyond monadic expansions. We also exhibit a paradigmatic structure for the failure of each of monadic NFCP/stable/NIP and prove each of these paradigms definably embeds into a monadic expansion of a sufficiently saturated model of any theory without the corresponding property.

15 pages; to appear in Model Theory

References in corpus (4)

• Characterizations of monadic NIP
• Uniformly bounded arrays and mutually algebraic structures
• A note on stability and NIP in one variable
• Mutual algebraicity and cellularity

Cited by in corpus (2)

• Mutual algebraicity and cellularity
• Theories with few non-algebraic types over models, and their decompositions