The automorphism group of the random graph: four conjugates good, three conjugates better. (Q1398268)

The random graph (or countable universal graph) \(R\) is the unique countable graph whose vertex set \(V(R)\) is the union of two sets \(P,Q\) such that \(P\cap Q=\emptyset\) and there exists a vertex \(u\in V(R)-(P\cup Q)\) adjacent to all vertices from \(P\) and to none from \(Q\). Let the automorphism group of \(R\) be \(G\). In 1995, the present author published the proof that if \(\sigma\) and \(\tau\) are non-identity elements of \(G\), then \(\tau\) may be written as the product of five conjugates of \(\sigma\). Then M. Rubin improved this number to four. In this paper, the author improves it to three.