  Let 0<n^*< omega and f:X-> n^*+1 be a function where X subseteq omega
backslash (n^*+1) is infinite. Consider the following set S_f= {x subset
aleph_omega : |x| <= aleph_{n^*} & (for all n in X)cf(x cap alpha_n)=
aleph_{f(n)}}. The question, first posed by Baumgartner, is whether S_f is
stationary in [alpha_omega]^{< aleph_{n^*+1}}. By a standard result, the above
question can also be rephrased as certain transfer property. Namely, S_f is
stationary iff for any structure A=< aleph_omega, ... > there's a B prec A such
that |B|= aleph_{n^*} and for all n in X we have cf(B cap aleph_n)=
aleph_{f(n)}. In this paper, we are going to prove a few results concerning the
above question.
