Maximal chains of isomorphic subgraphs of countable ultrahomogeneous graphs

Abstract

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.

Positive families and Boolean chains of copies of ultrahomogeneous structures

“…Theorem 3.2 in [14] guaranties that P = . Hence, Theorem 3.6(a) in [12] implies that there is a maximal chain L in 〈P(X), ⊂〉 isomorphic to L. Thus B R ⊂ M X . [3].…”

“…One is to present some new results about positive families. The other one is to provide a natural context for the recent research from [11][12][13]. For a countably infinite set X , a family P ⊂ P (X ) is called a positive family on X (see [10]) iff (P1) P ⊂ [X ] ω , (P2) P A ⊂ B ⊂ X ⇒ B ∈ P , (P3) A ∈ P ∧ |F | < ω ⇒ A\F ∈ P , (P4) ∃ A ∈ P |X \ A| = ω.…”

“…Let B R be the subclass of order types from C R for which the corresponding compact set K is, in addition, nowhere dense. Main results from [12,13] state that for a countable ultrahomogeneous partial order P…”

Antichains of Copies of Ultrahomogeneous Structures

“…Third, in [7,8,9] a classification of relational structures with respect to the properties of posets P(X), ⊂ is given. Fourth, the order types of the maximal chains in the posets of copies of countable ultrahomogeneous graphs and countable ultrahomogeneous partial orders are described in [10,11]. Finally, if X is a first order structure and R right Green's pre-order on its self-embedding monoid, Emb X, the corresponding antisymmetric quotient Emb X/ ≈ R , R (right Green's order) is isomorphic to the partial order P(X), ⊃ .…”

On the structure of random hypergraphs

“…Background and the statement of the result. We completely characterize chains of isomorphic substructures of the Fraïssé limit of finite n-uniform hypergraphs for each n > 1, thus generalizing some results from [8] and [7] to higher dimensions. Fraïssé theory, the systematic study of ultrahomogeneous universal structures, was initiated in the mid 1950's by Roland Fraïssé [2].…”