Let $H=(V,E)$ be a hypergraph. A set $M\subseteq E$ consisting of mutually disjoint members of $E$ is said to be a *matching*. We say $S\subseteq V$ is *matchable* if there is a matching $M$ such that $\bigcup M = S$.

One might think thatZorn's Lemma implies that every matchable set is contained in a maximal matchable set (with respect to $\subseteq$), but this is false: If $H = (\mathbb{N}, E)$ with $E = \big\{\{0,\ldots, n\}: n\in \mathbb{N}\big\}$, then $E$ is exactly the collection of matchable sets in $H$, and there are no maximal matchable sets in $H$.

**Question.** Let $n$ be a positive integer. If $H=(V,E)$ is a hypergraph such that every member of $E$ has at most $n$ elements, is there necessarily a maximal matchable set $S\subseteq V$?