## Coloring connected Hausdorff spaces

Motivation. I stumbled over the following hypergraph coloring concept when reading about an old (and open) problem by Erdos and Lovasz. Let $H=(V,E)$ be a hypergraph such that for all $e\in E$ we have $|e| > 1$, and let $Z \neq \emptyset$ be a set. Then a map $c:V\to Z$ is said to be a (hypergraph) coloring if for all $e\in E$ the restriction $c|_e$ is not constant (that is, the vertices pertaining to any edge are colored with more than 1 color). Trivially, for any nonempty hypergraph with no singleton edges, the identity map $\text{id}:V\to V$ is a coloring. The chromatic number of a hypergraph with no singleton edges is the least cardinal $\kappa$ such that there is a coloring $c:V\to \kappa$.

Equivalent topological formulation. If $(X,\tau)$ is a connected Hausdorff space such that $|X| > 1$, we can use this to color the associated hypergraph $(X,\tau\setminus\{\emptyset\})$. (Indeed we can apply it to any topological space without isolated points.)

We can reformulate this in topological terms in the following way: Let $(X,\tau)$ be a connected Hausdorff space. We define the nowhere dense covering number $\nu(X)$ to be the minimum cardinality of a set ${\cal N}$ of nowhere dense subsets of $X$ such that $\bigcup {\cal N} = X$. It is not hard to see that $\nu(X)$ equals the chromatic number of the hypergraph $(X,\tau\setminus\{\emptyset\})$.

For many standard connected Hausdorff spaces $X$ we have $\nu(X) = 2$. For instance, if $X = \mathbb{R}$ with the Euclidean topology, let ${\cal N} = \{\mathbb{Q}, \mathbb{R}\setminus\mathbb{Q}\}$. It took some effort to see that there are connected Hausdorff spaces $X$ with $\nu(X) > 2$.

Question. For which cardinals $\kappa > 2$ is there a connected Hausdorff space $(X,\tau)$ such that $\nu(X) = \kappa$?

Many more natural questions arise in this context, such as how does $\nu(\cdot)$ behave with topological products, and so on. So far I haven’t found a reference studying this concept of “Hausdorff space coloring”.