Failure on n-uniqueness: a family of examples

In this paper, the connections between model theory and the theory of infinite permutation groups are used to study the n-existence and the n-uniqueness for n-amalgamation problems of stable theories. We show that, for any n>1, there exists a stable theory having (k+1)-existence and k-uniqueness, for every k<n+1, but that does not have neither (n+2)-existence nor (n+1)-uniqueness. In particular, this generalizes the example, for n=2, due to E.Hrushovski given in [3].