{"paper":{"title":"Copies of the Random Graph: the 2-localization","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Milo\\v{s} S. Kurili\\'c, Stevo Todor\\v{c}evi\\'c","submitted_at":"2014-11-12T11:23:51Z","abstract_excerpt":"Let $G$ be a countable graph containing a copy of the countable random graph (Erd\\H{o}s-R\\'enyi graph, Rado graph), $Emb (G)$ the monoid of its self-embeddings, ${\\mathbb P} (G)=\\{f[G]: f\\in Emb (G)\\}$ the set of copies of $G$ contained in $G$, and ${\\mathcal I}_G$ the ideal of subsets of $G$ which do not contain a copy of $G$. We show that the poset $< {\\mathbb P} (G), \\subset>$, the algebra $P (G)/{\\mathcal I}_G$, and the inverse of the right Green's pre-order $< Emb (G),\\preceq ^R >$ have the 2-localization property. The Boolean completions of these pre-orders are isomorphic and satisfy the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.3144","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}