I'm hoping some of you might be able to help me with this question and hint me twords a solution using the probabilistic method.
Let P be a set of at most $2^{k/3}$ points in the plane. Prove that there exists a coloring of P with two colors such that in every open disc that contains at least k points both colors are present.
I've tried bounding the number of discs with $\binom{n}{3}$ for n the number of points, but i've failed to find a compelling argument.
thanks