Extensive approach to absolute homogeneity

TL;DR

This paper investigates absolutely homogeneous structures, especially metric spaces and edge-colored graphs, establishing a categorical framework, classifying ultrametric spaces and certain edge-colored graphs, and exploring products of such graphs.

Contribution

It introduces a categorical approach to absolutely homogeneous objects, establishes a correspondence with classes of finite objects, and classifies specific homogeneous metric spaces and graphs.

Findings

Full classification of absolutely homogeneous ultrametric spaces.

Classification of edge-colored graphs with isosceles or tricolored triangles.

Development of a product construction for edge-colored graphs.

Abstract

The main aim of the paper is to study in greater detail absolutely homogeneous structures (that is, objects with the property that each partial isomorphism extends to a global automorphism), with special emphasis on metric spaces and (possibly infinite, full) graphs with edge-coloring. Besides, a general categorical approach to this concept is presented. The main achievement of the paper is the discovery of one-to-one correspondence between absolutely homogeneous objects and certain classes (that become sets when isomorphic objects are identified) of "finite" objects that satisfy a few quite general axioms (such as amalgamation and heredity). It is also introduced and discussed in detail the concept of products for graphs with edge-coloring (that produces an absolutely homogeneous graph provided all factors are so). Among the most significant results of the paper, it is worth mentioning a full classification (up to isometry) of all absolutely homogeneous ultrametric spaces as well as of all absolutely homogeneous graphs with edge-coloring in which all triangles are isosceles or in which all triangles are (precisely) tricolored.