Matching number in infinite hypergraphs If $H= (V,E)$ is a hypergraph, a matching is a set $M\subseteq E$ such that $e_1\cap e_2 = \emptyset$ whenever $e_1\neq e_2 \in M$. The matching number $\mu(H)$ of a hypergraph $H=(V,E)$ with $V$ finite is the maximum number of elements a matching can have.
For infinite hypergraphs $H=(V,E)$, we let $$\mu(H) = \sup\{|M|:M\subseteq E\text{ is a matching}\}.$$ This definition agrees with the above definition for finite hypergraphs.
Question. If $H=(V,E)$ is a hypergraph with $V$ infinite, is there necessarily a matching $M_0\subseteq E$ such that $|M_0|=\mu(H)$?
 A: I think if all edges are finite, then there is such a matching.
Let $M_0$ be a maximal matching (in the sense that no edge can be added to it; this exists by Zorn's Lemma). Let $\alpha_0$ be the cardinality of $M_0$, and let $V_0$ be the set of all vertices contained in some edge of $M_0$. By maximality of $M_0$, every edge contains some vertex in $V_0$. Since no two edges of any matching contain the same vertex in $V_0$, we conclude that $|M| \leq |V_0|$ for every matching $M$.
If $\alpha_0$ is finite, then so is $|V_0|$ and thus $|V_0|$ is a finite upper bound on the size of any matching. This clearly implies that there is a matching of size $\mu(H) < \infty$.
If $\alpha_0$ is infinite, then $|V_0| = \alpha_0$ (because all edges are finite). So any matching has cardinality at most $\alpha_0$ thus showing that $\mu(H) \leq \alpha_0 = |M_0| \leq \mu(H)$.
A: Let $(P,\leq)$ be a poset, let $H$ be the set of maximal chains in $P$ —- so $H$ will be the set of vertices. For $a\in P$, let $E_a$ denote the set of all chains in $H$ containing $a$; these sets are the edges.
Now, $E_a$ and $E_b$ are disjoint iff $a$ and $b$ are incomparable. So the edges are pairwise disjoint iff the corresponding elements form an antichain.
It remains to find a poset containing arbitrarily large antichains but having no infinite ones. One such example is $\mathbb Z_{\geq0}^2$ with  a componentwise relation.
This provides a counterexample, if infinite edges are allowed.
