Problem 2
A configuration of points in the plane is called Colombian if it consists of red points and 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: (1) no line passes through any point of the configuration; (2) no region contains points of both colours. Find the least value of such that for any Colombian configuration of points, there is a good arrangement of lines.
Step 4 of 5: Verify that every region is monochromatic
In plain words
The isolated point or pair is monochromatic, while every region left outside the strips contains only the other colour.
Detailed analysis
Choose the auxiliary lines generically and close enough that none passes through a point. Each isolated strip contains exactly its same-colour pair (or the single hull point), and every point outside all strips has the other colour. Thus no region contains both colours, proving that 2013 lines suffice.