MathDB

There are n points in the plane with the property that the distance between any two points is at least 1. Prove that for sufficiently large n , the number of pairs of points whose distance is in

[ t_1 , t_1 + 1] \cup [ t_2 , t_2 + 1]

for some

t_1, t_2

, is at most

[\frac{n^2}{3}]

, and the bound is sharp.

Problem Statement