The [De Bruijn–Erdős theorem](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(graph_theory))  states that when all finite subgraphs of a graph $G$ can be colored with $n$ colors, the same is true for the whole graph.


There is a natural notion of coloring for [hypergraphs](https://en.wikipedia.org/wiki/Hypergraph) which is as follows. Let $H= (V, E)$ be a hypergraph, and let $\kappa\neq \emptyset$ be a cardinal. Then a map $c:V\to\kappa$ is said to be a *(hypergraph) coloring* if the restriction $c\restriction\_e : e \to \kappa$ is non-constant whenever $e$ has more than $1$ element.


Is the following statement true?



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> Let $n>1$ be an integer, and let $H=(V,E)$ be a hypergraph such that for all finite $E\_0\subseteq E$, the hypergraph $(V,E\_0)$ can be colored with $n$ colors. Then $H$ can be colored with $n$ colors.
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De Bruijn–Erdős theorem for hypergraphs
The space $X=\{1,\dots,n\}^V$ of all colorings (proper or not) of $H=(V,E)$ with $n$ colors is compact in the product topology. Given a finite set $F \subset E$, the set $K\_{F}$ of proper colorings of $(V,F)$ is a closed set in $X$.


For any finite collection
$F\_1,\dots,F\_k$ of finite subsets of $E$, the intersection
$$\cap\_{j=1}^k K\_{F\_j}=K\_{\cup\_{j=1}^k F\_j}$$ is nonempty by the given hypothesis.
Therefore, by compactness of $X$, the intersection
$$\bigcap \{K\_F : \, F \; \text{finite}, \; \, F \subset E \} $$
is nonempty, and any coloring in this intersection is a proper coloring of $H=(V,E)$.


