The famous De Bruijn–Erdős theorem and its hypergraphs generalization states the following.

**Theorem.** *Let $V$ be a set, and $E\subset2^V$ be a family of its subsets. Assume that every $e \in E$ is finite and that for some $k \in \mathbb{N}$, every finite sub-hypergraph of $H=(V,E)$ can be properly colored with $k$ colors. Then $H$ can also be properly colored with $k$ colors.*

It looks like that the standard proof based on Tychonoff's theorem doesn't work when the edges are allowed to be infinite. So, I wonder if the following analogue of De Bruijn–Erdős theorem holds.

Let $V$ be a set, and $E\subset2^V$ be a family of its subsets. Assume that every $e \in E$ is

countable. Moreover, assume that for some $k \in \mathbb{N}$ and for everycountable$W \subset V$, the hypergraph $H(W)= (W,E(W))$ can be properly colored with $k$ colors, where $E(W) = \{e \in E: e \subset W\}$. Then $H=(V,E)$ can also be properly colored with $k$ colors.

Erdős and Hajnal proved in [EH] the following related result on families of bounded intersections.

**Theorem.** *Let $V$ be a set, and $E\subset2^V$ be a family of its subsets. Assume that every $e \in E$ is countably infinite. Moreover, assume that there exists $m \in \mathbb{N}$ such that $|e_1\cap e_2|<m$ for all $e_1,e_2 \in E$. Then $H=(V,E)$ can be properly colored with $2$ colors.*

Can this result be strengthened as follows to deal with all families of finite intersections?

Let $V$ be a set, and $E\subset2^V$ be a family of its subsets. Assume that every $e \in E$ is

countably infinite, and that $e_1\cap e_2$ is finite for all $e_1,e_2 \in E$. Then $H=(V,E)$ can be properly colored with $2$ colors.

[EH] P. Erdős & A. Hajnal, *On a property of families of sets*, Acta Math. Academiae Scientiarum Hungarica, 12, 87–123 (1964).