  Recall that the Rado graph is the unique countable graph that realizes all
one-point extensions of its finite subgraphs. The Rado graph is well-known to
be universal and homogeneous in the sense that every isomorphism between finite
subgraphs of $R$ extends to an automorphism of $R$.
  We construct a graph of the smallest uncountable cardinality $\omega_1$ which
has the same extension property as $R$, yet its group of automorphisms is
trvial. We also present a similar, although technically more complicated,
construction of a complete metric space of density $\omega_1$, having the
extension property like the Urysohn space, yet again its group of isometries is
trivial. This improves a recent result of Bielas.
