MathNet 2014 Question 80

A set of lines in the plane is in general position if no two are parallel and no three pass through the same point. Such lines divide the plane into regions; those of finite area are called finite regions. Prove that for every sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt n$ of them blue so that no finite region has all its boundary sides blue.



Supplemental illustration of an admissible configuration; not an extra hypothesis.