For $n > 1$, a configuration consists of $2n$ distinct points in a plane, $n$ of them red, the remaining $n$ blue, with no three points collinear. A pairing consists of $n$ line segments, each with one blue and one red endpoint, such that each of the given $2n$ points is an endpoint of exactly one segment. Prove the following. a) For any configuration, there is a pairing in which no two of the $n$ segments intersect. (Hint: consider total length of segments.) b) Given $n$ red points (no three collinear), we can place $n$ blue points such that any pairing in the resulting configuration will have two segments that do not intersect. (Hint: First consider the case $n = 2$.)
For $n > 1$, a configuration consists of $2n$ distinct points in a plane, $n$ of them red, the remaining $n$ blue, with no three points collinear. A pairing consists of $n$ line segments, each with one blue and one red endpoint, such that each of the given $2n$ points is an endpoint of exactly one segment. Prove the following.\\
a) For any configuration, there is a pairing in which no two of the $n$ segments intersect. (Hint: consider total length of segments.)\\
b) Given $n$ red points (no three collinear), we can place $n$ blue points such that any pairing in the resulting configuration will have two segments that do not intersect. (Hint: First consider the case $n = 2$.)