A *hypergraph* $H=(V,E)$ consists of a
set $V$ and $E\subseteq {\mathcal P}(V)$. If $S\subseteq V$, we define
$$E|_S = \{e\cap S: (e\in E) \land (e\cap S \neq \emptyset)\}$$
and call $(S, E|_S)$ the *induced sub-hypergraph* of $H$.

We say that $H=(V,E)$ is *connected*
if for all $X \subseteq V$ with $\emptyset \neq X \neq V$ there
is $e\in E$ such that $$e\cap X \neq \emptyset \neq e
\cap (V\setminus X).$$

Let $H=(V,E)$ be a hypergraph and 
$\kappa\neq \emptyset$ be a cardinal.
Then a map $c:V\to \kappa$ is said to be a *colouring* if for
every $e\in E$ with $|e|\geq 2$ we have that
the restriction $c\restriction_e$ is non-constant. The *chromatic
number* $\chi(H)$ of $H$ is the smallest cardinal $\kappa$ such there
is a colouring $c:V\to \kappa$.

If $H=(V,E)$ is a hypergraph
and $S_1, S_2\subseteq V$ are disjoint, we say they are *connected
to each other* if there is $e\in E$ such that $$e\cap S_1 \neq \emptyset
\neq e\cap S_2.$$

**A form of Hadwiger's conjecture for hypergraphs.** Assume that $H=(V,E)$ is a hypergraph and let's assume 
$V\neq \emptyset \neq E$ to avoid pathologies. Let $\kappa \neq \emptyset$ be a
cardinal such that there is **no** colouring $c:V\to \kappa$. 

Then there is a collection ${\mathcal S}$ of non-empty, mutually
disjoint subsets of $V$ with $|{\mathcal S}| = \kappa$ such that

1. $(S, E|_S)$ is a connected hypergraph for each 
		$S\in {\mathcal S}$, and 
2. whenever $S\neq T\in {\mathcal S}$ then $S, T$ are connected
	to each other.

(In the graph context, this amounts to saying that there
is a complete minor of cardinality $\kappa$.)

**Question.** Does this version of Hadwiger's conjecture hold for hypergraphs with infinite chromatic number? 

(A similar statement is [known to hold for graphs with infinite chromatic number](https://arxiv.org/abs/1312.2829), but it is open whether it is valid for graphs with finite chromatic number.)