  Given a metric space $(X,d)$, we say that a mapping $\chi:
[X]^{2}\longrightarrow\{0.1\}$ is an isometric coloring if $d(x,y)=d(z,t)$
implies $\chi(\{x,y\})=\chi(\{z,t\})$. A free ultrafilter $\mathcal{U}$ on an
infinite metric space $(X,d)$ is called metrically Ramsey if, for every
isometric coloring $\chi$ of $[X]^{2}$, there is a member $U\in\mathcal{U}$
such that the set $[U]^{2}$ is $\chi$-monochrome. We prove that each infinite
ultrametric space $(X,d)$ has a countable subset $Y$ such that each free
ultrafilter $\mathcal{U}$ on $X$ satisfying $Y\in\mathcal{U}$ is metrically
Ramsey. On the other hand, it is an open question whether every metrically
Ramsey ultrafilter on the natural numbers $\mathbb{N}$ with the metric $|x-y|$
is a Ramsey ultrafilter. We prove that every metrically Ramsey ultrafilter
$\mathcal{U}$ on $\mathbb{N}$ has a member with no arithmetic progression of
length 2, and if $\mathcal{U}$ has a thin member then there is a mapping
$f:\mathbb{N}\longrightarrow\omega $ such that $f(\mathcal{U})$ is a Ramsey
ultrafilter.
