infinite combinatorics – Strongly Minimal Covers for Clique Hypergraphs of Graphs Answer

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infinite combinatorics – Strongly Minimal Covers for Clique Hypergraphs of Graphs

A hypergraph $H$ is a pair consisting of a set $V$ of vertices and a family of subsets of $V$ called edge. One class of examples is obtained by taking a graph $G=(V,E)$ with set of vertices $V$ and set of edges $E$ where $H=(V,Cliq (G) )$. Here $e\in Cliq(G)$ if and only if for all $v,w\in e$ look for that $v\ne w$ we have $\{v,w\}\in E$.

If $H=(V,F)$ is a hypergraph, a covers is a family $C$ of subsets of $F$ look for that $\bigcup C=V$. The cover is strongly minimal if for any cover $D$, $|C\setminus D|\le|D\setminus C|$.

Please give an example of a hypergraph stemming from the cliques of a graph as above that has no strongly minimal cover.

A related post with a nice example by domotorp is here.

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