Let denote the Rado graph and
its automorphism group. We show that any action of
on
by automorphisms which has finitely many orbits is isomorphic to the natural action. Therefore, in a sense, every interpretation of the Rado graph in itself without parameters is trivial. The proof uses a counting argument. As a corollary, every interpretation of the Rado graph in itself without parameters is definably isomorphic or anti-isomorphic to the Rado graph. Also, there is no interpretation of
– the Rado graph with a constant – in
which does not use parameters.
Tag Archives: random graph
Least supports and non-interpretability
In this post, we give a proof that does not interpret (without parameters) in the homogeneous poset
nor in the random graph
. The idea is to first show that every continuous action of
or
on a set is faithful or trivial. We show this by using the fact that
and
have least supports. Since
and
have an automorphism of order two and
does not, this proves that there is no nontrivial continuous action of
nor
on
.