We present explicit constructions of universal homogeneous objects in categories of domains
with stable embedding–projection pairs as arrows. These results make use of a
representation of such domains through graph-like structures and apply a generalization of
Rado’s result on the existence of the universal homogeneous countable graph. In particular,
we build universal homogeneous objects in the categories of coherence spaces and
qualitative domains, introduced by Girard (Girard 1987; Girard 1986), and two categories
of hypercoherences recently studied by Ehrhard (Ehrhard 1993). Our constructions rely on
basic numerical notions. We also show that a suitable random construction of Rado’s graph
and its generalizations produces with probability 1 the universal homogeneous structures
presented here.