  A group A acting faithfully on a set X is 2-distinguishable if there is a
2-coloring of X that is not preserved by any nonidentity element of A,
equivalently, if there is a proper subset of X with trivial setwise stabilizer.
The motion of an element a in A is the number of points of X that are moved by
a, and the motion of the group A is the minimal motion of its nonidentity
elements. For finite A, the Motion Lemma says that if the motion of A is large
enough (specifically at least 2 log_2 |A|), then the action is
2-distinguishable. For many situations where X has a combinatorial or algebraic
structure, the Motion Lemma implies the action of Aut(X) on X is
2-distinguishable in all but finitely many instances.
  We prove an infinitary version of the Motion Lemma for countably infinite
permutation groups, which states that infinite motion is large enough to
guarantee 2-distinguishability. From this we deduce a number of results,
including the fact that every locally finite, connected graph whose
automorphism group is countably infinite is 2-distinguishable. One cannot
extend the Motion Lemma to uncountable permutation groups, but nonetheless we
prove that 2-distinguishable permutation groups with infinite motion are dense
in the class of groups with infinite motion. We conjecture an extension of the
Motion Lemma which we expect holds for a restricted class of uncountable
permutation groups, and we conclude with a list of open questions. The
consequences of our results are drawn for orbit equivalence of infinite
permutation groups.

Combinatorics