Take any set of 2n points in the plane with no three collinear, and then arbitrarily color each point red or blue. Prove that it is always possible to pair up the red points with the blue points by drawing line segments connecting them so that no two of the line segments intersect.
This problem is provided in Richard A. Brauldi's book on Introductory Combinatorics. I believe it is implied that the number of red points is equal to the number of blue points. I tried to reason as follows: it seems like it's always possible to divide the plane into two regions, such that each region contains an equal number of red and blue points. Further, subdivide those regions into subregions with equal numbers of red and blue points; and continue the process until all the regions contain exactly one red and one blue point. Then we can simply connect the two points of every region with a line; and since all pairs lie in distinct regions, it seems obvious that no lines will intersect.
However, I don't know how exactly to prove this. That is, how do I gaurantee that it is in fact possible to divide any region of red and blue points into subregions of equal numbers of red and blue points? Or how would the problem be approached by a more professional. Thank you in advance.