Ranks Based on Strong Amalgamation Fraisse Classes

TL;DR

This paper introduces the concept of K-rank for types in model theory, based on strong amalgamation Fraisse classes, and explores its properties and relationships with existing ranks like dp-rank and op-dimension.

Contribution

It defines K-rank for strong amalgamation Fraisse classes and analyzes its properties and connections to other model-theoretic ranks.

Findings

K-rank can be computed for linear orders, equivalence relations, and graphs.

K-rank relates to dp-rank and op-dimension, providing new insights into model-theoretic complexity.

The paper establishes foundational properties of K-rank in various classes.

Abstract

In this paper, we introduce the notion of K-rank, where K is an strong amalgamation Fraisse class. Roughly speaking, the K-rank of a partial type is the number of "copies" of K that can be "independently coded" inside of the type. We study K-rank for specific examples of K, including linear orders, equivalence relations, and graphs. We discuss the relationship of 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).