  The Tarski number of an action of a group G on a set X is the minimal number
of pieces in a paradoxical decomposition of it. For any k>3 we construct a
faithful transitive action of a free group of rank k-1 with Tarski number k.
Using similar techniques we construct a group action of a free group F with
Tarski number 6 such that the Tarski numbers of restrictions of this action to
finite index subgroups of F are arbitrarily large.
