In this paper a hypergraph will be identified with the family of its edges. A hypergraph $\mathcal E$ possesses property $C(k,ρ)$ iff $|\bigcap \mathcal E'|<ρ$ 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 $λ$ and $ρ$ are cardinals, $S$ is a set of cardinals, $k\in ω$, then we write $$\mathbf M({λ},{S},{k},{μ})\to \mathbf{MinVC} $$ iff every hypergraph $\mathcal E\subset {[λ]}^{S}$ possessing property $C({k},{ρ})$ has a minimal vertex cover. If $S=\{κ\}$, then we simply write $\mathbf M({λ},κ,{k},{μ})\to \mathbf{MinVC}$ for $\mathbf M({λ},\{κ\},{k},{μ})\to \mathbf{MinVC}$ A set $S$ of cardinals is "nowhere stationary" iff $S\cap α$ is not stationary in $α$ for any ordinal $α$ with $cf(α)>ω$. Countable sets of cardinals, and sets of successor cardinals are nowhere stationary. In this paper we prove: (1) $\mathbf M({λ},{S},{2},{k})\to \mathbf{MinVC}$ for each nowhere stationary set $S$ of cardinals and $ω\le λ$, (2) $\mathbf M({λ},{κ} ,{2},{ρ})\to \mathbf{MinVC}$ provided $ρ<\beth_ω\le κ\le λ$, (3) $\mathbf M({λ},{ω},{r},{k})\to \mathbf{MinVC}$ provided $ω\le λ$ and $k,r\in ω$, (4) $\mathbf M({λ},{ω_1},{3},{k})\to \mathbf{MinVC}$ provided $ω_1\le λ$ and $k\in ω$.