# Strong and weak chromatic number of infinite bounded hypergraphs

This is a follow-up of an older question.

Let $H = (V, E)$ be a hypergraph such that every member of $E$ has more than $1$ element. Let $\kappa$ be a cardinal. We say that $c: V\to \kappa$ is a weak coloring if for all $e \in E$ the restriction $c|_e$ is not constant. We call $c$ a strong coloring, if for all $e \in E$ the restriction $c|_e$ is injective. We let $\chi_w(H)$ be the smallest cardinal such that there is a weak coloring to that cardinal, and we define $\chi_s(H)$ similarly for strong colorings.

Suppose that $V$ is infinite and that there is an integer $n\in\mathbb{N}$ such that $1<|e|\leq n$ for all $e\in E$. Is it possible that $\chi_w(H)$ is finite, but $\chi_s(H)$ is infinite?

Of course. Let $V$ be a set of non-zero integers, and $E$ consists of triples $(a,b,c)$ such that not all $a,b,c$ have the same sign. That is, the sign is a weak 2-colouring. On the other hand, for a strong colouring all elements must have different colours, since each pair of vertices is contained in a triple from $E$.