# Maximizing set systems with property $\mathbf{B}$

Let $$X$$ be an infinite set, and let $${\cal E}$$ be a collection of non-empty subsets of $$X$$. We say that $${\cal E}$$ has property $$\mathbf{B}$$ if there is $$B\subseteq X$$ such that $$B\cap E\neq \emptyset$$ and $$E\not\subseteq B$$ for all $$E\in{\cal E}$$. (This is equivalent to saying that the hypergraph $$(X,{\cal E})$$ has chromatic number $$2$$.)

It is possible to find a chain $${\frak E}$$ (with respect to $$\subseteq$$) of collections with property $$\mathbf{B}$$ such that $$\bigcup {\frak E}$$ has not property $${\mathbf B}$$ - but this does not imply that not every set of subsets of $$X$$ with property $${\mathbf B}$$ is not in a maximal such set of subsets. This leads to the following:

Question. What is an example of a set $$X$$ and and set of non-empty subsets $${\cal E}$$ of $$X$$ with property $${\mathbf B}$$ such that $${\cal E}$$ is not contained in a set of non-empty subsets $${\cal E'}$$ of $$X$$ that is maximal with respect to having property $${\mathbf B}$$ and set inclusion?

Any collection $$\cal E$$ satisfying property $${\mathbf B}$$ (further: $${\mathbf B}$$-collection) is contained in an inclusion-maximal $${\mathbf B}$$-collection.
Proof: consider the corresponding 2-coloring of $$\cal E$$ and define the over-collection $$\cal E_0$$ as the collection of all sets which contain elements of both colors. It is easy to see that the 2-coloring for $$\cal E_0$$ is unique (sets of size 2 from $$\cal E_0$$ are enough for such a conclusion), therefore if we add any set to $$\cal E_0$$, it is no longer a $${\mathbf B}$$-collection.