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Let $S$ be a set of $n$ points in the plane, the distance between any two of which is at least one. Show that there are at most $3n$ pairs of points of S at distance exactly one.

Can this be done with a unit circle and we can place at max. $6$ points on the perimeter and doing the same for other points as well ? i.e. we can get $6n/2 = 3n$ pairs at max. ?