What is the maximum number of points in the Euclidean plane with the property that given any three points, at least two are at distance one apart?
A lower limit for N dimensions is found by a generalization of armando's suggestion: 2N+4.
This is described by any two distinct regular N-dimensional polygon formed by faces of equilateral triangles with edge length 1. So for N=2, this is an equilateral triangle, for N=3 a triangular pyramid, etc.
For any three points, two of three of the points have to be on the same polygon, and each point in each polygon is a distance of 1 from each other point in the polygon. So this solution gives us a lower bound for the answer in N dimensions: for two dimensions, this is 6, for 3 dimensions, this is 8.
Posted by Avin
on 2005-05-24 01:18:00