Worst case expansions of complete theories

TL;DR

This paper characterizes the worst-case complexity of expanding models of complete theories with subsets, identifying the largest classes of theories stable under various types of monadic expansions.

Contribution

It precisely determines the complexity bounds of monadic expansions and identifies monadically NFCP theories as the maximal class stable under broader expansions.

Findings

Monadic NFCP theories are the largest class stable under non-monadic expansions.

Explicit structures demonstrate the failure of stability, NIP, and NFCP properties in expansions.

Definable embeddings show the universality of these paradigms across 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.