# Maximal matchable set in hypergraph with finite edges

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 $$S\subseteq V$$?

• That's right @MikhailTikhomirov - I have corrected it just now! Thank you also to Peter, and apologies for my mistake. May 3, 2022 at 13:26

Let $$n = 2$$, and $$H$$ be the complete bipartite graph with halves $$V_1$$ a copy of $$\mathbb{N}$$ and $$V_2$$ a copy of $$\mathbb{R}$$. Let $$r(M)$$ be an element of $$V_2$$ not covered by a matching $$M$$. If $$M$$ doesn't cover an $$x \in V_1$$, extend $$M$$ with $$(x, r(M))$$, otherwise increment every $$V_1$$-endpoint in $$M$$, and add $$(1, r(M))$$. This shows no $$S$$ is maximal.