  Suppose L is a relational language and P in L is a unary predicate. If M is
an L-structure then P(M) is the L-structure formed as the substructure of M
with domain {a: M models P(a)}. Now suppose T is a complete first order theory
in L with infinite models. Following Hodges, we say that T is relatively
lambda-categorical if whenever M, N models T, P(M)=P(N), |P(M)|= lambda then
there is an isomorphism i:M-> N which is the identity on P(M). T is relatively
categorical if it is relatively lambda-categorical for every lambda. The
question arises whether the relative lambda-categoricity of T for some lambda
>|T| implies that T is relatively categorical.
  In this paper, we provide an example, for every k>0, of a theory T_k and an
L_{omega_1 omega} sentence varphi_k so that T_k is relatively
aleph_n-categorical for n < k and varphi_k is aleph_n-categorical for n<k but
T_k is not relatively beth_k-categorical and varphi_k is not
beth_k-categorical.
