  In this paper a hypergraph will be identified with the family of its edges. A
hypergraph $\mathcal E$ possesses property $C(k,{\rho})$ iff $|\bigcap \mathcal
E'|<{\rho}$ for each $\mathcal E'\in {[\mathcal E]}^{k}$. A vertex set
$Y\subset \bigcup\mathcal E$ is a "vertex cover" of $\mathcal E$ iff $E\cap
Y\ne \emptyset$ for each $E\in \mathcal E$. A vertex cover $Y$ is "minimal" iff
no proper subset of $Y$ is vertex cover. If $A$ is a set and $S$ is a set of
cardinals, write $$ {[A]}^{S}=\{B\subset A: |B|\in S\}. $$ If ${\lambda}$ and
${\rho}$ are cardinals, $S$ is a set of cardinals, $k\in {\omega}$, then we
write $$\mathbf M({{\lambda}},{S},{k},{{\mu}})\to \mathbf{MinVC} $$ iff every
hypergraph $\mathcal E\subset {[{\lambda}]}^{S}$ possessing property
$C({k},{{\rho}})$ has a minimal vertex cover. If $S=\{\kappa\}$, then we simply
write $\mathbf M({{\lambda}},{\kappa},{k},{{\mu}})\to \mathbf{MinVC}$ for
$\mathbf M({{\lambda}},\{{\kappa}\},{k},{{\mu}})\to \mathbf{MinVC}$ A set $S$
of cardinals is "nowhere stationary" iff $S\cap {\alpha}$ is not stationary in
${\alpha}$ for any ordinal ${\alpha}$ with $cf({\alpha})>{\omega}$. Countable
sets of cardinals, and sets of successor cardinals are nowhere stationary. In
this paper we prove:
  (1) $\mathbf M({{\lambda}},{S},{2},{k})\to \mathbf{MinVC}$ for each nowhere
stationary set $S$ of cardinals and ${\omega}\le {\lambda}$,
  (2) $\mathbf M({{\lambda}},{{\kappa}} ,{2},{{\rho}})\to \mathbf{MinVC}$
provided ${\rho}<\beth_{\omega}\le {\kappa}\le {\lambda}$,
  (3) $\mathbf M({{\lambda}},{{\omega}},{r},{k})\to \mathbf{MinVC}$ provided
${\omega}\le {\lambda}$ and $k,r\in {\omega}$,
  (4) $\mathbf M({{\lambda}},{{\omega}_1},{3},{k})\to \mathbf{MinVC}$ provided
${\omega}_1\le {\lambda}$ and $k\in {\omega}$.
