Let $X=P_f(\mathbb{N})$ be the set of all finite subsets of the set of natural numbers. Let $S^*$ be a binary relation over $X$, defined by: for each $A,B\in X$, $\langle A,B \rangle \in S^*$ if and only if $A\cap B=\emptyset$.
Let $L$ be a vocabulary which contains binary relations $S$ and $R$, and contains a constant symbol $c_A$ for each $A\in X$.
Now, let $M$ be a structure to interpret $L$, defined by:
- the domain is $X$, $S^M=S^*$,
- $R^M$ is defined so that for each $A,B\in X$, $\langle A,B \rangle \in R^M$ if and only if $A\subseteq B$, and
- $c_A^{M}=A$ for each $A\in X$.
In addition, for each $n\in \mathbb{N}$, let $A_n=\{0,1,2,...,n-1\}$ (and $A_0=\emptyset$).
Let $T$ be the complete theory of $M$. We define $T_1$, $T_2$, and $T_3$ in $L^+=L \cup \{d\}$, where $d$ is a new constant symbol: \begin{align*} T_1 &= T \cup \lbrace S(c_{A_n},d)\land R(c_{\lbrace n,n+1,...,2n\rbrace},d)\mid n\in \mathbb{N}\rbrace\\ T_2&=T \cup \lbrace (S(c_{\lbrace n+1 \rbrace},d)\rightarrow R(c_{\lbrace n\rbrace},d)\mid n\in \mathbb{N}\rbrace\\ T_3&=T \cup \lbrace (S(c_{\lbrace 2n \rbrace},d)\land R(c_{\lbrace 2n+1\rbrace},d)\mid n\in \mathbb{N}\rbrace \end{align*}
- Prove that $T_1$ is inconsistent.
- Prove that $T_2$ is consistent, and it has a model which is an expansion of $M$ to $L^+$.
- Prove that $T_3$ is consistent, but it does not have a model which is an expansion of $M$ to $L^+$.
The formulas $R(c_A,x)\to\lnot S(c_A,x)$ are valid in $M$ for every nonempty $A$, and so are $R(c_B,x)\to R(c_A,x)$ and $S(c_B,x)\to S(c_A,x)$ whenever $A\subseteq B$.
So, $T_1$ implies that both $S(c_{\{1\}},d)$ and $R(c_{\{1\}},d)$ holds (taking the formula for $n=1,2$), which is a contradiction.
and 3. You can verify that for every finite subset of the given formulas, one can find interpretation of $d$ that satisfies those. So, you can use compactness to conclude. However, a $d\in X$ in neither cases exists which satisfies all the formulas at once, as it would have infinite many elements.