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?

(In reply to Bryan's solution by Bryan)

Right!  I forgot about the planar restriction.

Bravo, Bryan!  Your solution is rock solid.  All points not at distance 1 from a given point form an equilateral triangle of side 1 or a segment of length 1.  I'll be really surprised if anyone can do better.

Your figure reminds me of the problem of coloring the points in the plane with as few colors as possible so that no two points at distance 1 have the same color.  Your figure shows that at least 4 colors are required.

Posted by McWorter on 2005-05-24 05:19:28