A hypergraph is a pair $H=(V,E)$ such that $V$ is a (possibly infinite) set and $E\subseteq \mathcal{P}(V)$. $C\subseteq E$ is said to be a *cover* if $\bigcup C = V$ and $C$is *minimal* if $C'\subseteq C$ and $C'\neq C$ imply $\bigcup C'\neq V$. 

We call $H=(V,E)$ a *flag complex* if the following conditions are met:

  1. $e\in E, e'\subseteq E$ implies $e'\in E$;
  2. $\bigcup E = V$;
  3. $H$ is *2-determined*, that is if $S\subseteq V$ and for all $s, t \in S$ we have $\{s,t\}\in E$ then $S\in E$.

A standard application of Zorn's Lemma shows that in a flag complex, every edge $e\in E$ is contained in a maximal edge $m\in E$ ($m$ being maximal in $E$ with respect to set inclusion). We denote the collection of maximal edges by $\text{Max}(E)$.

**Question**: Is there a flag complex $H=(V,E)$ and a cover $M\subseteq \text{Max}(E)$ such that for every cover $M'\subseteq M$ we have that $M'$ is not mimimal?