Question 2
A configuration of $4027$ points in the plane is called *Colombian* if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is *good* for a Colombian configuration if the following two conditions are satisfied:
(i) No line passes through any point of the configuration.
(ii) No region contains points of both colors.
Find the least value of $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines.
(i) No line passes through any point of the configuration.
(ii) No region contains points of both colors.
Find the least value of $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines.