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?

Take a regular pentagon with side lengths 1 in the xy-plane.  Add points on both sides of this plane so that each is distance 1 from the vertices of the regular pentagon.

For any 3 points on the regular pentagon, at least two are adjacent, whence they are at distance 1.  If one of the 3 points is off the pentagon, then at least one of the 3 is on the pentagon, whence those two are at distance 1.  So the maximum number of points is at least 7, as Bryan suggests.

Posted by McWorter on 2005-05-24 00:42:24