Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). We call $H$ *proper* if $E\neq\emptyset, \emptyset \notin E$ and for no $e\_1\neq e\_2\in E$ we have $e\_1\subseteq e\_2$. A *matching* is a set $M$ of pairwise disjoint edges (members of $E$), and $T\subseteq V$ is said to be a *transversal* if $T\cap e \neq \emptyset$ for all $e\in E$.


It is easy to see that any transversal has at least the cardinality of any matching.


Given infinite cardinals $\alpha < \beta$ is there a proper hypergraph $H=(V,E)$ with the following properties?


1. there is a matching of size $\alpha$, and no matching has cardinality larger than $\alpha$, and
2. $\beta$ is the minimum size of a transversal of $H$.



Sizes of matchings and transversals in hypergraphs
Yes. For fixed $\beta$ we construct the hypergraph $G\_\beta$ with edges $\{e\_x\}\_{x\in \beta}$, any two edges $e\_x,e\_y$ have unique common vertex $v\_{x,y}$, and there are no other vertices. In this hypergraph the maximal matching has size 1 and the minimal transversal has size $\beta$, since any set consisting of $\gamma<\beta$ vertices intersects at most $2\gamma<\beta$ edges. 


Now take $\alpha$ disjoint copies of $G\_\beta$.


