minimum number of unit distances required for a unit equilateral triangle

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Problem. Suppose we have $n$ points on the plane. Among $\binom{n}{2}$ pairwise distances, there are $e$ number of unit distances. Find minimum $e$ ($e$ as a function of $n$) so that there is a equilateral triangle with unit sides.

The problem is a generalization of Mexico 1995 National Olympiad Problem 2 posted on AOPS here