Publication date:
7 September 2017
Source:Advances in Mathematics, Volume 317
Author(s): Miloš S. Kurilić, Stevo Todorčević
Let $〈R,\sim 〉$ be the Rado graph, $\mathrm{Emb}(R)$ the monoid of its self-embeddings, $\mathbb{P}(R)=\{f(R):f\in \mathrm{Emb}(R)\}$ 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 $〈\mathbb{P}(R),\subset 〉$, the algebra $P(R)/{\mathcal{I}}_R$, and the inverse of the right Green's preorder on $\mathrm{Emb}(R)$, and show that these preorders are forcing equivalent to a two step iteration of the form $\mathbb{P}⁎\pi$, where the poset $\mathbb{P}$ is similar to the Sacks perfect set forcing: adds a generic real, has the ${\mathrm{\aleph }}_0$-covering property and, hence, preserves ${\omega }_1$, 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.