Distinguishing number of Urysohn metric spaces

Abstract

The distinguishing number of a structure is the smallest size of a partition of its elements so that only the trivial automorphism of the structure preserves each cell of the partition. We show that for any countable subset of the positive real numbers, the corresponding countable homogeneous Urysohn metric space, when it exists, has distinguishing number 2 or the distinguishing number is infinite. While it is known that a sufficiently large finite primitive structure has distinguishing number 2, unless its automorphism group is the full symmetric group or alternating group, the infinite case is open and these countable Urysohn metric spaces provide further confirmation toward the conjecture that all primitive homogeneous countably infinite structures have distinguishing number 2 or else the distinguishing number is infinite.