Let $H=(V,E)$ be a [hypergraph](https://en.wikipedia.org/wiki/Hypergraph). We say that $I\subseteq V$ is an *independent set* if $e\not\subseteq I$ for all $e\in E$. 

We say that $H$ is *tameable* if every independent set is contained in a maximal independent set. Every graph [is tameable](https://mathoverflow.net/a/385522/8628) and, more generally, so is every hypergraph with finite edges.

There are easy examples of non-tameable hypergraphs, and I use this one given by user @bof in the comment section of [this answer](https://mathoverflow.net/a/385522/8628): Let $H=(\omega,[\omega]^\omega)$, where $[\omega]^\omega$ denotes the collection of infinite subsets of $\omega$. (The only independent subsets of this graph are the finite sets, and there is no maximal finite set.)

If $(P,\leq)$ is a poset, then with $\text{Max}(P)$ we denote the collection of maximal elements of $P$. (Note that $\text{Max}(\omega) = \varnothing$, for instance.)

Given a hypergraph $H=(V,E)$ we let $$\text{Tame}(H) = \{E'\subseteq E: (V, E') \text{ is tame}\}.$$

**Question.** Given a hypergraph $H=(V,E)$ with $V\neq\varnothing\neq E$ and $\varnothing\notin E$, do we necessarily have $\text{Max}(\text{Tame}(H)) \neq \varnothing$?