### Cofinalities of elementary substructures of structures on $\aleph_\omega$

by Liu and Shelah. [LiSh:484]

Israel J Math, 1997

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.

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