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Copies of the Random Graph | Miloš S. Kurilić
; Stevo Todorčević
; | Date: |
23 Oct 2014 | Abstract: | Let $(R, sim )$ 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 | Services: | Forum | Review | PDF | Favorites |
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