Minimal covers of hypergraphs

Abstract

For a hypergraph $H=(V,\mathcal{E})$, a subfamily $\mathcal{C}\subseteq \mathcal{E}$ is called a cover of the hypergraph if $\bigcup \mathcal{C}=\bigcup \mathcal{E}$. A cover $\mathcal{C}$ is called minimal if each cover $\mathcal{D}\subseteq \mathcal{C}$ of the hypergraph $H$ coincides with $\mathcal{C}$. 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 $(\omega ,\omega )$. This characterization implies that a countable hypergraph $(V,\mathcal{E})$ has a minimal cover if every infinite set $I\subseteq V$ contains a finite subset $F\subseteq I$ such that the family of edges ${\mathcal{E}}_F:=\{E\in \mathcal{E}:F\subseteq E\}$ is finite. Also we prove that a hypergraph $(V,\mathcal{E})$ has a minimal cover if $\sup \{|E|:E\in \mathcal{E}\}<\omega$ or for every $v\in V$ the family ${\mathcal{E}}_v:=\{E\in \mathcal{E}:v\in E\}$ is finite.

5 pages