  We introduce the notion of echeloned spaces - an order-theoretic abstraction
of metric spaces. The first step is to characterize metrizable echeloned
spaces. It turns out that morphisms between metrizable echeloned spaces are
uniformly continuous or have a uniformly discrete image. In particular, every
automorphism of a metrizable echeloned space is uniformly continuous, and for
every metric space with midpoints the automorphisms of the induced echeloned
space are precisely the dilations.
  Next we focus on finite echeloned spaces. They form a Fraisse class and we
describe its Fraisse-limit both as the echeloned space induced by a certain
homogeneous metric space and as the result of a random construction. Building
on this we show that the class of finite ordered echeloned spaces is Ramsey.
The proof of this result combines a combinatorial argument by Nesetril and
Hubicka with a topological-dynamical point of view due to Kechris, Pestov, and
Todorcevic. Finally, using the method of Katetov functors due to Kubis and
Masulovic, we prove that the full symmetric group on a countable set
topologically embeds into the automorphism group of the countable universal
homogeneous echeloned space.
