Let $H = (V, E)$ be a [hypergraph](https://en.wikipedia.org/wiki/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 all $e\in E$ have finitely many elements. (I.e., $H$ has *finite rank*(\*).) We do not require that $H$ be uniform. Is it possible that $\chi\_w(H) < \aleph\_0$ but $\chi\_s(H) \geq \aleph\_0$?


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(\*) Cf. e.g. page 2 in [C. Berge, *Hypergraphs*, North Holland, 1989, ISBN 0 444 87489 5].



Strong and weak chromatic number of infinite hypergraphs of finite rank
Since the OP did not require that the rank of the hypergraph be *bounded*, the **answer to the question is 'obviously yes'**. An example is any hypergraph $H$ which is(\*) the 



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> disjoint union of $\aleph\_0$-many hyperedges of *increasing cardinality*. 
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Then $\chi\_w(H)=2$ but $\chi\_s(H)=\aleph\_0$. This answers the question.


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(\*)Formally,let $\{v\_{i,j}\colon (i,j)\in\omega^2\}$ be a set of $\aleph\_0$-many pairwise distinct indeterminates, and let $H:=(\omega,\{\{ v\_{i,j}\mid\ j\in i \}\mid\ i\in\omega\})$.


