Minimal covers of hypergraphs

TL;DR

This paper characterizes when hypergraphs have minimal covers, linking this property to the absence of certain subhypergraphs and providing conditions related to the finiteness of edges and neighborhoods.

Contribution

It provides a complete characterization of hypergraphs with minimal covers, connecting it to the structure of subhypergraphs and finiteness conditions.

Findings

Equivalence of conditions for the existence of minimal covers in hypergraphs.

Characterization involving the absence of the hypergraph $( ext{omega}, ext{omega})$.

Conditions ensuring minimal covers based on edge size and vertex neighborhoods.

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.