# Sizes of matchings and transversals in hypergraphs

Let $$H=(V,E)$$ be a 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 $$H$$.

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 $$G_\beta$$.