Maximal Chains of Isomorphic Subgraphs of Countable Ultrahomogeneous Graphs

TL;DR

This paper characterizes the order types of maximal chains of isomorphic subgraphs in countable ultrahomogeneous graphs, linking them to specific classes of compact sets of reals.

Contribution

It provides a complete characterization of the order types of maximal chains in P(G) for various classes of countable ultrahomogeneous graphs.

Findings

Order types correspond to compact sets of reals with specific properties.

For Rado and Henson graphs, order types are those of compact sets with a minimal non-isolated point.

For unions of disjoint complete graphs, order types are those of compact nowhere dense sets with a minimal non-isolated point.

Abstract

For a countable ultrahomogeneous graph G let P(G) denote the collection of domains of subgraphs of G isomorphic to G. The order types of maximal chains in the set P(G) U {\o} ordered by the inclusion 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 H_n, for some n>2; (II) the order types of compact nowhere dense sets of reals having the minimum non-isolated, if G is the union of \mu disjoint complete graphs of size \nu, where \mu \nu =\omega .