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 McWorter)

McWorter's proposal is valid except for the requirement that the points be coplanar.

My solution with 7 points is to start with diamond ABCD made of two adjacent equilateral triangles with sides of unit length:

  A
B  D 
  C

Any two points on this diamond will be 1 unit apart, except for A and C, so we will consider that case. 

Place an identical diamond AEFG on top of ABCD, then rotate AEFG about A until C and F are one unit apart.  The third point is either E or G, which is one unit from A, or it is F, which is one unit from C.  In any case, at least two of the three points are 1 unit apart. 

 

Posted by Bryan on 2005-05-24 03:14:20