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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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