Give an example of a structure of cardinality $\omega_2$ that has a substructure of $\omega$ but no substructure of $\omega_1$ 
Give an example of a structure of cardinality $\omega_2$ that has a substructure of $\omega$ but no substructure of $\omega_1$

This is from Hodges' A Shorter Model Theory. 
My idea is to take some set of cardinality $\omega_2$ as the domain, $\mathrm{dom}(A)$. Choose and fix a subset of $\mathrm{dom}(A)$ with cardinality $\omega$, call this $X$. I need to then choose an $n$-ary relation, $R^A$ carefully. The point is that I need to find something that breaks:
$R^A=R^B\cap A^n$
for every subset of cardinality $\omega_1$ but such that that holds for A subset of cardinality $\omega$. The ones I've tried (like everything in the set $X$ being related to each other, etc.) didn't work.
Any hints?

HINT: Using only relations won't work, since given a language with only relation symbols, and a structure to the language, every subset of the domain defines a substructure. So using functions is necessary here.
Let $\{f_\alpha\mid\alpha<\omega_2\}$ be a list of $\omega_2$ unary function symbols. For sake of concreteness we can take $A=\omega_2$. Find a way to interpret $f_\alpha$ so for every $n<\omega$, $f_\alpha(n)<\omega$; and if $\beta\geq\omega$, then $\{f_\alpha(\beta)\mid\alpha<\omega_2\}=\omega_2$ (you actually don't need equality there, requiring that $|\{f_\alpha(\beta)\mid\alpha<\omega_2\}|=\aleph_2$ is enough).
