Školjka

%V0 A set of line sin the plane is in [i]general position[/i] if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite area; we call these its [i]finite regions[/i]. Prove that for all sufficiently large $n$, in any set of $n$ lines in general position it is possible to colour at least $\sqrt{n}$ of the lines blue in such a way that none of its finite regions has a completely blue boundary. [i]Note:[/i] Results with $\sqrt{n}$ replaced by $c \sqrt{n}$ will be awarded points depending on the value of the constant $c$.