Copies of the Random Graph

TL;DR

This paper explores the structure of copies of the Rado graph within itself, showing that certain associated forcing notions are equivalent to a two-step iteration involving Sacks-like forcing, preserving key set-theoretic properties.

Contribution

It establishes the forcing equivalence of various posets related to the Rado graph's copies and embeddings to a two-step iteration involving Sacks-like forcing and an $\omega$-distributive forcing.

Findings

Posets related to the Rado graph are forcing equivalent to a two-step iteration.

The poset $P$ is similar to Sacks perfect set forcing, adding a generic real.

The Boolean completions of these posets are isomorphic.

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.