The following arose from a question in model theory (specifically in the model theory of modules).
Fix an arity $k$. Let $[\mathbb{Q}]^k$ denote the set of all subsets of $\mathbb{Q}$ of cardinality $k$. Let $X_k$ be the set of all finite subsets of $[\mathbb{Q}]^k$; i.e. $X_k$ is the set of all finite $k$-uniform hypergraphs with vertices from $\mathbb{Q}$.
$X_k$ is a group under symmetric difference, and as such it is in fact a vector space over $\mathbb{F}_2$, the field of two elements. For a basis we can take the hypergraphs with a single edge. Now $\mbox{Aut}(\mathbb{Q}, <)$ (the automorphism group of the ordered rationals) acts on $X_k$ by permuting the vertices. Given $H \in X_k$ let $\mathcal{O}(H)$ denote its orbit under $\mbox{Aut}(\mathbb{Q}, <)$. Say that $Y \leq X_k$ is invariant if whenever $H \in Y$, then also $\mathcal{O}(H) \subseteq Y$.
Question. What are the invariant subspaces of $X_k$?
This question is somewhat vague; to be more precise, we can instead ask: how many invariant subspaces of $X_k$ are there? (I expect finitely many.)
As an example, if $k = 0$ then $X_k$ has dimension one, and hence only has two subspaces (both invariant): $0$ and itself. If $X_k = 1$ then $X_k$ can be identified with finite subsets of the rationals, and it is not hard to check that there are precisely 3 invariant subspaces, namely $0$, $X_k$, and the set of all $H \in X_k$ with $|H|$ even.
