Points on a spherical surface with radius $4$ are colored in $4$ different colors. Prove that there exist two points with the same color such that the distance between them is either $4\sqrt{3}$ or $2\sqrt{6}$. (Distance is Euclidean, that is, the length of the straight segment between the points)
Problem
Source: Spain Mathematical Olympiad 2018 P4
Tags: geometry, Spain, combinatorics, pigeonhole principle