Some weak indivisibility results in ultrahomogeneous metric spaces

TL;DR

This paper investigates the weak indivisibility property in ultrahomogeneous metric spaces, specifically the integer and rational Urysohn spaces, and compares it with age-indivisibility, providing new examples and insights.

Contribution

It offers new results on weak indivisibility in specific ultrahomogeneous metric spaces and compares it with age-indivisibility, including an example distinguishing the two.

Findings

Weak indivisibility does not always hold in the integer and rational Urysohn spaces.

A countable ultrahomogeneous metric space can be age-indivisible but not weakly indivisible.

Abstract

We study the validity of a partition property known as weak indivisibility for the integer and the rational Urysohn metric spaces. We also compare weak indivisiblity to another partition property, called age-indivisibility, and provide an example of a countable ultrahomogeneous metric space which may be age-indivisible but not weakly indivisible.