This question is motivated by considerations on [conflict-free colorings](https://mathoverflow.net/questions/416793/conflict-free-coloring-of-mathbbr-with-the-euclidean-topology), which arose while studying assignment problems for frequencies in cellular networks.


A *[hypergraph](https://en.wikipedia.org/wiki/Hypergraph)* $H=(V,E)$ is said to be *linear* if for all $e\_1\neq e\_2 \in E$ we have $|e\_1\cap e\_2| \leq 1$.


Let $H=(\omega, E)$ be a linear hypergraph such that $e$ is infinite for all $e\in E$. Is there a necessarily a map $c: \omega\to \{0,1\}$ such that for all $e\in E$ there is $v^\*\in e$ such that $$c^{-1}\big(\big\{c(v^\*)\big\}\big) \cap e \;= \;\{v^\*\}\;\;?$$


(More informally, we want every edge $e\in E$ to have a vertex $v^\*\in e$ such that all the vertices in $e\setminus\{v^\*\}$ are colored with the (unique) color in $\{0,1\}\setminus\{c(v^\*)\}$.)



Conflict-free coloring of linear hypergraphs on $\omega$
Let $V$ be the set of points in the affine plane $\mathbb Q^2$, and $E$ be the set of lines.


Assume that the coloring is possible. Say a line is *white* if it contains a unique black point, and *black* otherwise.


There exist two white points (on two parallel lines), hence a white line through them. There exists a white point outside that line, hence infinitely many white lines concurrent at that white point (and passing through white points on the found line). Similarly, take infinitely many black lines concurrent at a black point.


Take 5 white and 5 black lines out of those collections; they have at least 20 points of mutual intersection. But at most 5 of those points are black (one per white line), and at most 5 are black. A contradiction.


