Minimal vertex covers in infinite hypergraphs

A vertex cover of a hypergraph is set vertices which intersects each hyperedge. possesses property C(k,ρ) iff |⋂E′|<ρ for k element E′ hyperedges. Komjáth proved that every uniform possessing C(2,r) some r∈ω has minimal cover. In this paper we will relax the assumption uniformity to an cardinalities hyperedges “small” infinite cardinals, e.g. it countable, or does not contain uncountably many limit cardinals. also GCH decide following statement: If G C(2,ω) μ-uniform μ≥ω1, then Using Shelah's Revised theorem, show if strengthen μ≥ω1 μ≥ℶω, can prove statement in ZFC! We all are countably infinite, instead C(k,r) (for k∈ω) enough guarantee existence hyperedge cardinality ω1, only C(3,r) enough.