Asymmetric colouring of locally compact permutation groups

Abstract

AbstractLet be a group of permutations of a countable set . Call a colouring of asymmetric if no preserves all colours. The motion of is the minimal number of elements moved by an element . We show that if is locally compact with respect to the permutation topology and the motion of is infinite, then there is an asymmetric 2‐colouring of . This builds on a recent result by Babai, generalises this result, and confirms a conjecture by Imrich, Smith, Tucker, and Watkins from 2015.