Remarks on relative categoricity

TL;DR

This paper surveys the concept of relative categoricity, presents new results including a proof of the Gaifman conjecture for a special case, and discusses stability properties in model theory.

Contribution

It introduces new results on relative categoricity, including a proof of the Gaifman conjecture for almost internal covers, and discusses stability over P.

Findings

Gaifman conjecture proved for almost internal covers

Full omega-stability over P introduced

Counterexample provided to full stability over P

Abstract

The paper is partly a survey with historical background and references, partly provides the opportunity to put in print some unpublished early work, and partly has new results. A special case of relative categoricity is identified (almost internal covers) for which the Gaifman conjecture is proved, full omega-stability over P is introduced, and as counterexample is given to full stability over P.