Description
(Joint work work Tamás Csernák.)
A "vertex cover" of a hypergraph is a set of vertices which intersects each hyperedge.
A hypergraph possesses "property $C({k},{\rho})$" iff $|\bigcap \mathcal E'|<{\rho}$ for each $k$ element set $\mathcal E'$ of hyperedges.
Komjáth proved that every uniform hypergraph possessing property $C({2},{r})$ for some $r\in {\omega}$ has a minimal vertex cover.
We could relax the assumption of uniformity to an assumption that the set of cardinalities of the hyperedges is a ``small'' set of infinite cardinals,
e.g. it is countable, or it does not contain uncountably many limit cardinals.
Komjáth also proved that GCH does not decide the following statement:
"If a hypergraph $G$ possessing property $C({2},{{\omega}})$ is ${\mu}$-uniform for some ${\mu}\ge {\omega}_1$, then $G$ has a minimal vertex cover."
Using Shelah's Revised GCH theorem, we could show that if we strengthen the assumption ${\mu}\ge {\omega}_1$
to ${\mu}\ge beth_{\omega}$, then we can prove the statement in ZFC!
We also show that if all the hyperedges of a hypergraph are countably infinite,
then instead of $C({2},{r})$ the assumption $C({k},{r})$ (for some $k\in {\omega}$) is enough
to guarantee the existence of a minimal vertex cover.
If every hyperedge has cardinality ${\omega}_1$, then we can only prove that $C({3},{r})$ is enough.