Copies of the Random Graph

Abstract

Let $(R,\sim )$ be the Rado graph, $Emb(R)$ the monoid of its self-embeddings, $\Pi (R)=\{f[R]:f\in Emb(R)\}$ the set of copies of $R$ contained in $R$, and ${\mathcal{I}}_R$ the ideal of subsets of $R$ which do not contain a copy of $R$. We consider the poset $(\Pi (R),\subset )$, the algebra $P(R)/{\mathcal{I}}_R$, and the inverse of the right Green's pre-order on $Emb(R)$, and show that these pre-orders are forcing equivalent to a two step iteration of the form $P\ast \pi$, where the poset $P$ is similar to the Sacks perfect set forcing: adds a generic real, has the ${\aleph }_0$-covering property and, hence, preserves ${\omega }_1$, has the Sacks property and does not produce splitting reals, while $\pi$ codes an $\omega$-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.

arXiv comment28 pages