Towards the Classification of Polymorphism-homogeneous Metric Spaces
Bojana Pantić and Maja Pech
We explore the model-theoretic concept of polymorphism-homogeneity, a natural generalisation of the classical homogeneity, for certain classes of metric spaces, viewed as relational structures in an obvious way. At first, we prove that ultrametric spaces are polymorphismhomogeneous. Subsequently, we proceed to a thorough investigation of the so-called skeletal metric spaces, with respect to the diameter of their skeleton, a specific unique graph corresponding to them. Consequently, we obtain a characterization of metrically polymorphism-homogeneous connected graphs of diameter at most 3.
Keywords: Metric space, homogeneous, homomorphism homogeneous, polymorphism homogeneous, graphs