Ranks based on strong amalgamation Fraïssé classes

Abstract

In this paper, we introduce the notion of \({\textbf{K}} \)-rank, where \({\textbf{K}} \) is a strong amalgamation Fraïssé class. Roughly speaking, the \({\textbf{K}} \)-rank of a partial type is the number “copies” of \({\textbf{K}} \) that can be “independently coded” inside of the type. We study \({\textbf{K}} \)-rank for specific examples of \({\textbf{K}} \), including linear orders, equivalence relations, and graphs. We discuss the relationship of \({\textbf{K}} \)-rank to other ranks in model theory, including dp-rank and op-dimension (a notion coined by the first author and C. D. Hill in previous work).