Prove that there exists a set S of 3^{1000} points in the plane such that for each point P in S, there are at least 2000 points in S whose distance to P is exactly 1 inch.
Prove that there exists a set S of 3^{1000} points in the plane such that for each point P in S, there are at least 2000 points in S whose distance to P is exactly 1 inch.