Two-distance extremal configurations in the plane

Prove that, for all sufficiently large \(n\), the maximum total number \(f(n)=m_1(S)+m_2(S)\) of occurrences of the smallest and second smallest distances among \(n\)-point sets \(S\) in the plane equals the explicit lattice bound \(e(n)\), and classify all extremal point sets as sets similar to the extremal subsets of the triangular lattice described in Theorem \(\ref{the:e}\).

Background

The paper relates its positive edge-isoperimetric result for the triangular lattice Cayley graph $\Lambda_U$ to a problem of Erdős and Vesztergombi concerning occurrences of the two smallest distances determined by a finite set of points in the plane.

For an $n$-point set $S$, $m_1(S)$ and $m_2(S)$ count occurrences of the smallest and second smallest distances, respectively, and $f(n)$ is the maximum of their sum. Theorem $\ref{the:e}$ supplies a lower bound $f(n)\geq e(n)$ from subsets of the triangular lattice. The stated conjecture asks whether this lower bound is sharp for sufficiently large $n$, and whether the triangular-lattice extremal configurations are the only extremizers up to similarity.