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Copies of the random graph. (English) Zbl 1423.03189
Summary: Let $$\langle R, \sim \rangle$$ be the Rado graph, $$\mathrm{Emb}(R)$$ the monoid of its self-embeddings, $$\mathbb{P}(R) = \{f(R) : f \in \mathrm{Emb}(R) \}$$ 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 $$\langle \mathbb{P}(R), \subset \rangle$$, the algebra $$P(R) / \mathcal{I}_R$$, and the inverse of the right Green’s preorder on $$\mathrm{Emb}(R)$$, and show that these preorders are forcing equivalent to a two step iteration of the form $$\mathbb{P} \ast \pi$$, where the poset $$\mathbb{P}$$ 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.

##### MSC:
 03E40 Other aspects of forcing and Boolean-valued models 05C80 Random graphs (graph-theoretic aspects) 03C15 Model theory of denumerable and separable structures 03C50 Models with special properties (saturated, rigid, etc.) 06A06 Partial orders, general 20M20 Semigroups of transformations, relations, partitions, etc.
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##### References:
 [1] Cameron, P. J., The random graph, (The Mathematics of Paul Erdös II, Algorithms Combin., vol. 14, (1997), Springer Berlin), 333-351 · Zbl 0864.05076 [2] Erdös, P.; Rényi, A., Asymmetric graphs, Acta Math. Acad. Sci. Hungar., 14, 295-315, (1963) · Zbl 0118.18901 [3] Fraïssé, R., Theory of relations, Studies in Logic and the Foundations of Mathematics, vol. 145, (2000), North-Holland Amsterdam, with an appendix by Norbert Sauer · Zbl 0593.04001 [4] Halpern, J. D.; Läuchli, H., A partition theorem, Trans. Amer. Math. Soc., 124, 360-367, (1966) · Zbl 0158.26902 [5] Hodges, W., Model theory, Encyclopedia of Mathematics and Its Applications, vol. 42, (1993), Cambridge University Press Cambridge [6] Jech, T., Multiple forcing, Cambridge Tracts in Mathematics, vol. 88, (1986), Cambridge University Press Cambridge · Zbl 0601.03019 [7] Jech, T., Set theory, Perspectives in Mathematical Logic, (1997), Springer-Verlag Berlin · Zbl 0882.03045 [8] Kurilić, M. S., From $$A_1$$ to $$D_5$$: towards a forcing-related classification of relational structures, J. Symbolic Logic, 79, 1, 279-295, (2014) · Zbl 1337.03042 [9] Kurilić, M. S., Posets of copies of countable scattered linear orders, Ann. Pure Appl. Logic, 165, 895-912, (2014) · Zbl 1297.06001 [10] Kurilić, M. S., Forcing with copies of countable ordinals, Proc. Amer. Math. Soc., 143, 4, 1771-1784, (2015) · Zbl 1386.03065 [11] Kurilić, M. S., Different similarities, Arch. Math. Logic, 54, 7-8, 839-859, (2015) · Zbl 1373.03049 [12] Kurilić, M. S.; Kuzeljević, B., Maximal chains of isomorphic subgraphs of the Rado graph, Acta Math. Hungar., 141, 1, 1-10, (2013) · Zbl 1313.05342 [13] Kurilić, M. S.; Marković, P., Maximal antichains of isomorphic subgraphs of the Rado graph, Filomat, 29, 9, 1919-1923, (2015) · Zbl 1464.05329 [14] Kurilić, M. S.; Todorčević, S., Forcing by non-scattered sets, Ann. Pure Appl. Logic, 163, 1299-1308, (2012) · Zbl 1250.03102 [15] Kurilić, M. S.; Todorčević, S., The poset of all copies of the random graph has the 2-localization property, Ann. Pure Appl. Logic, 167, 8, 649-662, (2016) · Zbl 1432.03059 [16] Rado, R., Universal graphs and universal functions, Acta Arith., 9, 331-340, (1964) · Zbl 0139.17303 [17] Todorčević, S., Introduction to Ramsey spaces, Annals of Mathematics Studies, vol. 174, (2010), Princeton University Press Princeton, NJ · Zbl 1205.05001
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