$\DeclareMathOperator\Cliq{Cliq}$A *hypergraph* $H$ is a pair consisting of a set $V$ of *vertices* and a family of subsets of $V$ called *edges*.
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 $v,w\in e$ such that $v\ne w$ we have $\{v,w\}\in E$.

If $H=(V,F)$ is a hypergraph, a *cover* is a subset $C$ of $F$ such 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.

Edit: I had an incorrect definition of "cover" before. Thanks to Mikhail Tikhomirov for leading me to look at it again.

anticliques(independent sets). It seems clear that everylocally finitegraph has a strongly minimal cover by anticliques. $\endgroup$4more comments