A rigid Urysohn-like metric space

TL;DR

This paper constructs uncountable structures, both a graph and a metric space, that have the extension properties of well-known universal structures but possess trivial automorphism groups, demonstrating rigidity.

Contribution

It introduces the first uncountable Urysohn-like metric space with a trivial isometry group, extending Bielas's recent results to larger cardinalities.

Findings

Constructed an uncountable graph with the extension property and trivial automorphism group.

Developed a similar uncountable metric space with the same properties.

Showed that such rigid structures exist at the uncountable level, contrasting with their countable counterparts.

Abstract

Recall that the Rado graph is the unique countable graph that realizes all one-point extensions of its finite subgraphs. The Rado graph is well-known to be universal and homogeneous in the sense that every isomorphism between finite subgraphs of $R$ extends to an automorphism of $R$. We construct a graph of the smallest uncountable cardinality $\omega_1$ which has the same extension property as $R$, yet its group of automorphisms is trvial. We also present a similar, although technically more complicated, construction of a complete metric space of density $\omega_1$, having the extension property like the Urysohn space, yet again its group of isometries is trivial. This improves a recent result of Bielas.