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.
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 .