Authors: Kurilić, Miloš
Todorčević, Stevo 
Title: The poset of all copies of the random graph has the 2-localization property
Journal: Annals of Pure and Applied Logic
Volume: 167
Issue: 8
First page: 649
Last page: 662
Issue Date: 1-Aug-2016
Rank: M22
ISSN: 0168-0072
DOI: 10.1016/j.apal.2016.04.001
Let G be a countable graph containing a copy of the countable universal and homogeneous graph, also known as the random graph. Let Emb(G) be the monoid of self-embeddings of G, P(G)=(f[G]:f∈Emb(G)) the set of copies of G contained in G, and IG the ideal of subsets of G which do not contain a copy of G. We show that the poset 〈P(G),⊂〉, the algebra P(G)/IG, and the inverse of the right Green's pre-order 〈Emb(G), ≤R〉 have the 2-localization property. The Boolean completions of these pre-orders are isomorphic and satisfy the following law: for each double sequence [bnm:〈n, m〉∈ω×ω] of elements of B denotes the set of all binary subtrees of the tree ω<ω..
Keywords: 2-localization | Countable random graph | Forcing | Isomorphic substructure | Right Green's pre-order | Self-embedding
Publisher: Elsevier
Project: Set Theory, Model Theory and Set-Theoretic Topology 
CNRS and NSERC, Grant 455916

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