Copies of the Random Graph
Milo\v{s} S. Kurili\'c, Stevo Todor\v{c}evi\'c

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 be the Rado graph, the monoid of its self-embeddings, the set of copies of contained in , and the ideal of subsets of which do not contain a copy of . We consider the poset , the algebra , and the inverse of the right Green's pre-order on , and show that these pre-orders are forcing equivalent to a two step iteration of the form , where the poset is similar to the Sacks perfect set forcing: adds a generic real, has the -covering property and, hence, preserves , has the Sacks property and does not produce splitting reals, while codes an -distributive forcing. Consequently, the Boolean completions of these four posets are isomorphic and the same holds for each countable graph containing a copy…
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