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 2 of 5: Isolate a hull point or a same-colour hull pair
In plain words
A line placed just inside a supporting line can cut off a hull point or a small hull edge without touching any configuration point.
Detailed analysis
For the upper bound, use the following construction. If the convex hull has a red vertex A, choose a line close to a supporting line at A so that A alone is on one side. If the hull has no red vertex, all hull vertices are blue; since there are more blue than red points, two consecutive hull vertices A,B are blue, and one line close to the hull edge AB isolates exactly A and B.