Hi math experts!

Let P be a set of at most $\displaystyle 2^\frac{k}{3}$ 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.

Here is what I got so far:

Let $\displaystyle A_d$ be the event where the disc d is monochromatic.

Clearly Pr($\displaystyle A_d$ )=$\displaystyle 2^{k-1}$

Now I want to show that the Pr($\displaystyle \bigcup _{d \in D} A_d$ ) < 1. (where $\displaystyle D$ is the set of all discs) and this will nail the problem.

How can I use the fact that the number of points is bounded to bound the number of discs?

Any help will be appreciated