A vertex cover of a hypergraph is a set of vertices which intersects each hyperedge.A hypergraph possesses propertyC(k,ρ) iff |⋂E′|<ρ for each k element set E′ of hyperedges.Komjáth proved that every uniform hypergraph possessing property C(2,r) for some r∈ω has a minimal vertex cover. In this paper we will 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 propertyC(2,ω)is μ-uniform for someμ≥ω1, then G has a minimal vertex cover.Using Shelah's Revised GCH theorem, we show that if we strengthen the assumption μ≥ω1 to μ≥ℶω, 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∈ω) is enough to guarantee the existence of a minimal vertex cover. If every hyperedge has cardinality ω1, then we can only prove that C(3,r) is enough.