Distinguishing Number of Countable Homogeneous Relational Structures

TL;DR

This paper investigates the distinguishing number of countable homogeneous relational structures, providing new computations for various structures and proposing a conjecture that most such structures have distinguishing number two or are infinite.

Contribution

It computes the distinguishing number for several classes of countable homogeneous structures and proposes a conjecture about their typical distinguishing number.

Findings

Most structures have distinguishing number two or infinite

Computed for various finite and countable structures including graphs and posets

Supports conjecture that primitive structures also follow this pattern

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. Albertson and Collins computed the distinguishing number for various finite graphs, and Imrich, Klav\v{z}ar and 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.