For a language with a binary predicate $R$, consider the following two models $\mathcal{M_1}$ and $\mathcal{M_2}$ defined as such:
$\mathcal{M_1} = (|\mathcal{M_1}|, R_0)$, with root point $r$, and finite branches from $r$ of all even lengths, that is, $|\mathcal{M_1}| = \{r\} \cup \{a_{n,k}|0<n\leq k, k$ even $\}$
$R^\mathcal{M_1} = R_0 = \{\langle a_{n,k},a_{n+1,k} \rangle | 0 < n<k,k$ even$ \} \cup \{ \langle r,a_{1,k} \rangle | k$ even $\}$
$\mathcal{M_2} = (|\mathcal{M_2}|, R_1)$, with root point $q$, and finite branches from $q$ of all odd lengths, that is, $|\mathcal{M_2}| = \{q\} \cup \{b_{n,k}|0<n\leq k, k$ odd$\}$
$R^\mathcal{M_2} = R_1 = \{\langle b_{n,k},b_{n+1,k} \rangle | 0 < n<k,k$ odd$ \} \cup \{ \langle q,b_{1,k} \rangle | k$ odd $\}$
Note that we can think of these models as graphs whose points are related by $R$.
i) First, how might I go about showing that for any finite subset $X \subseteq |\mathcal{M}_i|$ for $i = 1$ and $i = 2$, there is a subset $Y \subseteq |\mathcal{M}_{1-i}|$ such that there is an isomorphism from $\mathcal{M}'_i \simeq \mathcal{M}'_{1-i}$, where $\mathcal{M}'_i = (X,R'_i)$ and $R'_i$ is the restriction of $R_i$ to $X$ and similarly for $\mathcal{M}'_{1-i}$
ii) I would like to show that the Spoiler (or the player trying to prove that the two structures are not elementary equivalent) in the Ehrenfreucht-Fraisse game of perfect information of length 3, $EF_3(\mathcal{M}_0,\mathcal{M}_1)$ has a winning strategy and understand what it is.
Background on this problem: In EF games, there are two players: the Spoiler and the Duplicator. First the Duplicator (who tries to show that the two models are elementary equivalent) selects a node in either model. Next the Spoiler (who tries to show that the models are not elementary equivalent) picks a node in the opposite model (including one which has already been picked). The two players continue to alternate turns until the designated number of turns have been taken. The nodes picked in either model form a substructure for each respective model denoted $\mathcal{M}'$ and if these two substructures are isomorphic, then the Duplicator wins and otherwise the spoiler wins.
iii) Third, I would like to find some sentence $\varphi$ such that $\vDash_{\mathcal{M}_0} \varphi$ but $\nvDash_{\mathcal{M}_1}$ as we know from part ii) that such a sentence exists.
Here are my thoughts (any feedback would be appreciated):
i) I suppose that we would use the result of compactness on these two structures, first trying to show they are isomorphic and the rest would follow. Is this strategy on the right track?
ii) Since we have a game of perfect information of finite length, we know we know that there is such a strategy (and we are told that it is for the Duplicator, who plays first). Suppose the Duplicator first picks the node (1,2) in the odd structure $\mathcal{M}_1$ and then the Spoiler would pick the node (1,3) in the odd structure and so forth. In a length-three game, it seems that this strategy would work. Is this a correct approach?
iii) I'm sure how to approach this one. I suppose that it would be some universal statement since the Duplicator has a winning strategy but not sure what it would be. Any tips would be appreciated.
Thanks in advance!