The poset of all copies of the random graph has the 2-localization property

Abstract

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..