  In this paper we further study links between concentration of measure in
topological transformation groups, existence of fixed points, and Ramsey-type
theorems for metric spaces. We prove that whenever the group $\Iso(\U)$ of
isometries of Urysohn's universal complete separable metric space $\mathbb U$,
equipped with the compact-open topology, acts upon an arbitrary compact space,
it has a fixed point. The same is true if $\U$ is replaced with any generalized
Urysohn metric space $U$ that is sufficiently homogeneous. Modulo a recent
theorem by Uspenskij that every topological group embeds into a topological
group of the form $\Iso(U)$, our result implies that every topological group
embeds into an extremely amenable group (one admitting an invariant
multiplicative mean on bounded right uniformly continuous functions). By way of
the proof, we show that every topological group is approximated by finite
groups in a certain weak sense. Our technique also results in a new proof of
the extreme amenability (fixed point on compacta property) for infinite
orthogonal groups. Going in the opposite direction, we deduce some Ramsey-type
theorems for metric subspaces of Hilbert spaces and for spherical metric spaces
from existing results on extreme amenability of infinite unitary groups and
groups of isometries of Hilbert spaces.
