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The Rado graph admits only one oligomorphic action of its automorphism group

Let R denote the Rado graph and \aut R its automorphism group. We show that any action of \aut R on R 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 (R,c) – the Rado graph with a constant – in R which does not use parameters.

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