Maximal chains of isomorphic subgraphs of the Rado graph

Kurilić, Miloš S.; Kuzeljević, Boriša

doi:10.1007/s10474-013-0341-9

Cited by 4 publications

(9 citation statements)

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“…(P2) holds because p ∈ P and q ∈ a∈Gp {a, a − 1, a + 1}. Thus p 1 ∈ P. Since H ⊂ G p ⊂ G p 1 and since, by (7) we have {q, k} ∈ ρ p 1 , for all k ∈ K, and {q, h} ∈ ρ p 1 , for all h ∈ H \ K, it follows that which implies that H is K n−1 -free and, hence, H, H ∈ C n (C, ρ C ). Suppose that b ∈ C H H .…”

Section: Maximal Chains Of Copies Of H Nmentioning

confidence: 87%

“…A contradiction to the choice of q. q = a + 1. Then by (8) we have b = q and, since a = q, by (7) we have {a, b} ∈ ρ p which implies a ∈ G p . A contradiction to the choice of q. q = b.…”

Section: Maximal Chains Of Copies Of H Nmentioning

confidence: 97%

“…e-mail: milos@dmi.uns.ac.rs 2 Mathematical Institute of the Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11001 Belgrade, Serbia. e-mail: borisa@mi.sanu.ac.rs P(X), ⊂ , where X is a relational structure, the class of order types of maximal chains in the poset P(G Rado ), ⊂ was characterized in [7]. The aim of this paper is to complete the picture for all countable ultrahomogeneous graphs in this context and, thus, the following theorem is our main result.…”

Section: Introductionmentioning

confidence: 98%

“…The statement (I) for the Rado graph is proved in [7] and a proof for the graphs H n is given in Section 4, while (II) is proved in Section 5.…”

Section: Introductionmentioning

confidence: 99%

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Maximal chains of isomorphic subgraphs of countable ultrahomogeneous graphs

Kurilić

Kuzeljević

2014

Advances in Mathematics

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For a countable ultrahomogeneous graph G = G, ρ let P(G) denote the collection of sets A ⊂ G such that A, ρ ∩ [A] 2 ∼ = G. The order types of maximal chains in the poset P(G) ∪ {∅}, ⊂ are characterized as:(I) the order types of compact sets of reals having the minimum nonisolated, if G is the Rado graph or the Henson graph H n , for some n ≥ 3;(II) the order types of compact nowhere dense sets of reals having the minimum non-isolated, if G is the union of µ disjoint complete graphs of size ν, where µν = ω. 2010 MSC: 05C63, 05C80, 05C60, 06A05, 06A06, 03C50, 03C15.

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Section: Maximal Chains Of Copies Of H Nmentioning

confidence: 87%

Section: Maximal Chains Of Copies Of H Nmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 98%

“…The statement (I) for the Rado graph is proved in [7] and a proof for the graphs H n is given in Section 4, while (II) is proved in Section 5.…”

Section: Introductionmentioning

confidence: 99%

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Maximal chains of isomorphic subgraphs of countable ultrahomogeneous graphs

Kurilić

Kuzeljević

2014

Advances in Mathematics

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“…Concerning the order theoretic properties of the poset P(R), ⊂ we note that it is a homogeneous, atomless and chain complete suborder of the order [R] ω , ⊂ having a largest element, R. In addition, by [14], it contains maximal antichains of size c, ℵ 0 and n, for each positive integer n, and in [13] the order types of maximal chains in this poset are characterized as the order types of sets of the form K \ {min K}, where K is a compact subset of the real line having the minimum non-isolated. The sets R H K (the orbits of R) will play an important role in our constructions.…”

Section: Fact 21 Let R ∼ Be a Rado Graph And P(r) The Set Of Its Comentioning

confidence: 99%

Copies of the random graph

Kurilić

Todorčević

2017

Advances in Mathematics

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Let R, ∼ be the Rado graph, Emb(R) the monoid of its self-embeddings, P(R) = {f (R) : f ∈ Emb(R)} the set of copies of R contained in R, and I R the ideal of subsets of R which do not contain a copy of R. We consider the poset P(R), ⊂ , the algebra P (R)/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 * π, where the poset P is similar to the Sacks perfect set forcing: adds a generic real, has the ℵ 0 -covering property and, hence, preserves ω 1 , has the Sacks property and does not produce splitting reals, while π codes an ω-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. 2010 MSC: 05C80, 03C15, 03C50, 03E40, 06A06, 20M20. Theorem 1.2 For each countable non-scattered graph G, ∼ and, in particular, for the Rado graph, the poset P(G) is forcing equivalent to the two-step iteration P * π, where 1 P "π is an ω-distributive forcing" and the poset P is similar to the Sacks forcing: adds a generic real, has the ℵ 0 -covering property (thus preserves ω 1 ), has the Sacks property and does not produce splitting reals. In addition, ro sq P(G) ∼ = ro(P (R)/I R ) + ∼ = ro(P * π) and these complete Boolean algebras are weakly distributive 3 .In fact, if G, ∼ is a countable graph containing a copy of the Rado graph, then these two structures are equimorphic and, by [11], forcing equivalent. So it is sufficient to prove the previous theorem assuming that G, ∼ is the Rado graph.Finally we note that the results of this paper are related to the investigation of the monoids of self-embeddings. We recall that the right Green's pre-order R on a monoid M, ·, 1 is defined by x R y iff x · z = y, for some z. It is easy to check (see [12]) that the poset of copies P(X) of a structure X is isomorphic to the antisymmetric quotient of the pre-order Emb(X), ( R ) −1 and, consequently, these pre-orders are forcing equivalent. Thus, by Theorem 1.2, for the Rado graph we have Emb(R), ( R ) −1 ≡ (P * π) and the Boolean completion of the preorder Emb(R), ( R ) −1 is a weakly distributive complete Boolean algebra. PreliminariesFirst we introduce a convenient notation. If G, ∼ is a graph (namely, if ∼ is a symmetric and irreflexive binary relation on the set G) and K ⊂ H ∈ [G] <ω , letThe object of our study is the Rado graph (the Erdős-Rényi graph, the countable random graph) introduced independently by Erdős and Rényi [2] and Rado [17]. 3 A complete Boolean algebra B is called weakly distributive (or (ω, ·, <ω)-distributive) iff for each cardinal κ and each matrix [bnα : n, α ∈ ω × κ] of elements of B we have n∈ω α∈κ bnα = s:ω→[κ] <ω n∈ω α∈s(n) bnα.

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Forcing with copies of the Rado and Henson graphs

Guzmán

Todorčević

2023

Annals of Pure and Applied Logic

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Maximal chains of isomorphic subgraphs of the Rado graph

Cited by 4 publications

References 7 publications

Maximal chains of isomorphic subgraphs of countable ultrahomogeneous graphs

Maximal chains of isomorphic subgraphs of countable ultrahomogeneous graphs

Copies of the random graph

Forcing with copies of the Rado and Henson graphs

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