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)∪{θ},∨〉 are characterized as:. (I) the order types of compact sets of reals having the minimum non-isolated, if G is the Rado graph or the Henson graph Hn, 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. © 2014 Elsevier Inc.