Distance sets of universal and Urysohn metric spaces

A metric space $\mathrm{M}=(M;\text{de})$ is {\em homogeneous} if for every isometry $f$ of a finite subspace of $\mathrm{M}$ to a subspace of $\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto $\mathrm{M}$ extending $f$. A metric space $\mathit{U}$ is an {\em Urysohn} metric space if it is homogeneous and separable and complete and if it isometrically embeds every separable metric space $\mathrm{M}$ with $\dist($\mathrm{M}$)\subseteq \dist(\boldsymbol{U})$\mathrm{.}(With$\dist($\mathrm{M}$)$ being the set of distances between points in $\mathrm{M}$.) The main results are: (1) A characterization of the sets $\text{dist}(\mathit{U})$ for Urysohn metric spaces $\mathit{U}$. (2) If $R$ is the distance set of an Urysohn metric space and $\mathrm{M}$ and $\mathrm{N}$ are two metric spaces, of any cardinality with distances in $R$, then they amalgamate disjointly to a metric space with distances in $R$. (3) The completion of a homogeneous separable metric space $\mathrm{M}$ which embeds isometrically every finite metric space $\mathrm{F}$ with $\text{dist}(\mathrm{F})\subseteq \text{dist}(\mathrm{M})$ is homogeneous.