Motivation. I stumbled over the following hypergraph coloring concept when reading about an old (and open) problem by Erdos and Lovasz. Let be a hypergraph such that for all we have , and let be a set. Then a map is said to be a (hypergraph) coloring if for all the restriction 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 is a coloring. The chromatic number of a hypergraph with no singleton edges is the least cardinal such that there is a coloring .
Equivalent topological formulation. If is a connected Hausdorff space such that , we can use this to color the associated hypergraph . (Indeed we can apply it to any topological space without isolated points.)
We can reformulate this in topological terms in the following way: Let be a connected Hausdorff space. We define the nowhere dense covering number to be the minimum cardinality of a set of nowhere dense subsets of such that . It is not hard to see that equals the chromatic number of the hypergraph .
For many standard connected Hausdorff spaces we have . For instance, if with the Euclidean topology, let . It took some effort to see that there are connected Hausdorff spaces with .
Question. For which cardinals is there a connected Hausdorff space such that ?
Many more natural questions arise in this context, such as how does behave with topological products, and so on. So far I haven’t found a reference studying this concept of “Hausdorff space coloring”.