VN-TST 1991 Problem 1

1.) In the plane let us consider a set $S$ consisting of $n \geq 3$ distinct points satisfying the following three conditions:

I. The distance between any two points $\in S$ is not greater than 1. II. For every point $A \in S$ , there are exactly two "neighbor" points, i.e. two points $X, Y \in S$ for which $AX = AY = 1$ . III. For arbitrary two points $A, B \in S$ , let $A', A''$ be the two neighbors of $A, B', B''$ the two neighbors of $B$ , then $A'AA'' = B'BB''$ .

Is there such a set $S$ if $n = 1991$ ? If $n = 2000$ ? Explain your answer.