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28 March 2024
 
  » arxiv » 1410.6320

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Copies of the Random Graph
Miloš S. Kurilić ; Stevo Todorčević ;
Date 23 Oct 2014
AbstractLet $(R, sim )$ be the Rado graph, $Emb (R)$ the monoid of its self-embeddings, $Pi (R)={ f[R]: fin Emb (R)}$ 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.
Source arXiv, 1410.6320
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