In this paper, assuming large cardinals, we prove the consistency of the following: Let $n \in \omega$ and $k_1, k_2 \leq n$. Let $f: \omega \rightarrow \{k_1, k_2\}$ be such that for all $n_1 < n_2 \in f^{-1}\{k_1\}, n_2 - n_1 \geq 4$. Then the set $S = \{x \subset \aleph_\omega:|x| = \omega_n \text{and} \forall m \geq n, cf(x \cap \omega_m) = \omega_{f(m)}\}$ is stationary in $\lbrack \aleph_\omega \rbrack^{<\omega_{n + 1}}$. The above is equivalent to the statement that for any structure $\mathscr{A}$ on $\aleph_\omega$, there is $\mathscr{B} \prec A$ such that $|\mathscr{B}| = \omega_n$ and for all $m > n, cf(\mathscr{B} \cap \omega_m) = \omega_{f(m)}$.