Sách Distinguishing Number of Countable Homogeneous Relational Structures

Abstract
The distinguishing number of a graph G is the smallest positive integer r such
that G has a labeling of its vertices with r labels for which there is no non-trivial
automorphism of G preserving these labels.
In early work, Michael Albertson and Karen Collins computed the distinguishing
number for various finite graphs, and more recently Wilfried Imrich, Sandi Klavˇzar
and Vladimir Trofimov computed the distinguishing number of some infinite graphs,
showing in particular that the Random Graph has distinguishing number 2.
We compute the distinguishing number of various other finite and countable
homogeneous structures, including undirected and directed graphs, and posets. We
show that this number is in most cases two or infinite, and besides a few exceptions
conjecture that this is so for all primitive homogeneous countable structures.