Minimal covers of hypergraphs
Taras Banakh, Dominic van der Zypen

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 , a subfamily is called a cover of the hypergraph if . A cover is called minimal if each cover of the hypergraph coincides with . We prove that for a hypergraph the following conditions are equivalent: (i) each countable subhypergraph of has a minimal cover; (ii) each non-empty subhypergraph of has a maximal edge; (iii) contains no isomorphic copy of the hypergraph . This characterization implies that a countable hypergraph has a minimal cover if every infinite set contains a finite subset such that the family of edges is finite. Also we prove that a hypergraph has a minimal cover…
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