Minimal covers of infinite hypergraphs

Abstract

For a hypergraph H=(V,E), a subfamily C⊆E is called a cover of the hypergraph if ⋃C=⋃E. A cover C is called minimal if each cover D⊆C of the hypergraph H coincides with C. We prove that for a hypergraph H the following conditions are equivalent: (i) each countable subhypergraph of H has a minimal cover; (ii) each non-empty subhypergraph of H has a maximal edge; (iii) H contains no isomorphic copy of the hypergraph (ω,ω). This characterization implies that a countable hypergraph (V,E) has a minimal cover if every infinite set I⊆V contains a finite subset F⊆I such that the family of edges EF≔{E∈E:F⊆E} is finite. Also we prove that a hypergraph (V,E) has a minimal cover if sup{|E|:E∈E}