Seventeen points have been picked in a plane, and each pair of points has been connected by a line segment of one of three colors: red, yellow, or green. Prove that there are three points which are the vertices of a monochrome triangle.
Monochrome triangle here means that all the sides of the triangle have the same color.
Source: The book, "A Moscow Math Circle" by Sergey Dorichenko.
I have seen the official answer given in the book. It is the same as the answer given by @KellyBundy below.
I want to see what other ways there are to solve this problem.