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?

(Inspired by Asaf Karagila)
Let $A$ be a structure with $\mathrm{dom}(A)=\omega_2$. $C^A=\emptyset$, $R^A=\emptyset$ and $F^A=\{f_{\alpha}\colon \alpha<\omega_2\}$ where:
$$f_{\alpha}(\beta)=\begin{cases}\beta \text{ for } \beta<\omega\\
\alpha \text{ for } \beta\geq \omega\end{cases}$$
Then define $S$ to be a substructure of $A$ with $\mathrm{dom}(S)=\omega$, $C^A=\emptyset$, $R^A=\emptyset$ and $F^A=\{f_{\alpha}\colon \alpha<\omega\}$. Then trivially $F^S=F^A\mid \mathrm{dom}(A)$ and $S$ is a substructure of $A$.
However if $T$ is a substructure with $|T|=\omega_1$, then there is some $t\in T\setminus\omega$. Pick $\alpha\in  A\setminus T$, then $f_{\alpha}(t)=\alpha\not\in T$.
