If $H= (V,E)$ is a [hypergraph](https://en.wikipedia.org/wiki/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 $|M\_0|=\mu(H)$?



Matching number in infinite hypergraphs
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.


