
Probabilistic Method
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($\displaystyle A_d$ )=$\displaystyle 2^{k1}$
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