BG-MO 1993 Problem 6

Find all natural numbers $n$ for which there exists set $S$ consisting of $n$ points in the plane, satisfying the condition: For each point $A \in S$ there exist at least three points say $X, Y, Z$ from $S$ such that the segments $AX, AY$ and $AZ$ have length $1$ (it means that $AX = AY = AZ = 1$ ).