We are only interested in partitions $S_1,S_2,...$ where each $S_i$ is nonempty and finite and $\bigcup S_i = S$ is countably infinite. A partition is computable (relative to the structure $\mathscr{A}$) if for each $n,i$ we can determine if $n \in S_i$ (given the atomic diagram of $\mathscr{A}$).
Observation:
Suppose that $V_{\infty}= \langle V, +\rangle$ is the $\omega$-dimensional vector space over a finite field of size $p$ for some prime $2<p<\omega$ defined pointwise such that every element $x \in V$ consists of almost all zeros. For any model of $V_{\infty}$ there is a partition given as follows
First consider $S = \{x \mid x \text{ is least nonzero element in the subspace generated by $x$}\}$
Then consider the set $\{ \overline{x} \mid x \in S \}$ where $\overline{x}= \{\text{the subspace generated by x}\}$ this is a computable partition of an infinite set relative to the atomic diagram of our model. Notice that for every $x \in S$, the size of the subspace it generates is $|\overline{x}| \leq p$ (I think almost all actually $=p$)
Question:
What if we consider that $V_\infty$ is the infinite vector space over field of size $2$, that is $F_2 = \{0,1\}$. If we consider the same $S$ and partition $\{\overline{x} \mid x \in S \}$ notice that almost every element $x \in S$ has $|\overline{x}|=1$ (every nonzero element is the only nonzero element in the subspace they generate). Are there any other partitions $S_1,S_2,...$ of finite sets (computable from the atomic diagram of our model) such that for most $i$ we have $|S_i|>1$ ? I can't seem to think of any others...