Copies of the random graph

Abstract

Let 〈R,∼〉 be the Rado graph, Emb(R) the monoid of its self-embeddings, P(R)={f(R):f∈Emb(R)} the set of copies of R contained in R, and IR the ideal of subsets of R which do not contain a copy of R. We consider the poset 〈P(R),⊂〉 the algebra P(R)/IR, and the inverse of the right Green's preorder on Emb(R), and show that these preorders are forcing equivalent to a two step iteration of the form P⁎π where the poset P is similar to the Sacks perfect set forcing: adds a generic real, has the ℵ0-covering property and, hence, preserves ω1, has the Sacks property and does not produce splitting reals, while π codes an ω-distributive forcing. Consequently, the Boolean completions of these four posets are isomorphic and the same holds for each countable graph containing a copy of the Rado graph.