  We consider random temporal graphs, a version of the classical
Erd\H{o}s--R\'enyi random graph G(n,p) where additionally, each edge has a
distinct random time stamp, and connectivity is constrained to sequences of
edges with increasing time stamps. We study the asymptotics for the distances
in such graphs, mostly in the regime of interest where np is of order log n. We
establish the first order asymptotics for the lengths of increasing paths: the
lengths of the shortest and longest paths between typical vertices, the maxima
of these lengths from a given vertex, as well as the maxima between any two
vertices; this covers the (temporal) diameter.
