# De Bruijn–Erdős theorem for hypergraphs

The De Bruijn–Erdős theorem 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 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?

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.

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 $$H=(V,E)$$.