Does anyone know if it has been proved what the maximum number of simplexes occurring in the plane is for a given value of $n$ points? I am interested in this question in relation to packing problems (eventually generalizing to tetrahedron packing in 3-space).
In 1974, Harborth proved the following theorem by applying Euler's polyhedral formula and using induction.
Let $f_{2}^{\text{min}}(n)$ denote the maximum number of times the minimum distance can occur among n points in the plane. Then, $f_{2}^{\text{min}}(n) = \lfloor 3n - \sqrt{12n -3} \rfloor$
For any dimension $d$, we have a general definition for the maximum number of times the minimum distance can occur among $n$ points in $\mathbb{R}^d$ as, $f_{d}^{\text{min}}(n) = \max_{|P|=n} | \{\overline{pq} | p,q \in P \text{ and } |p-q|=1\}|$ where $P \subseteq \mathbb{R}^d$ is any $n$-element point set. We now wish to extend this result to not only consider the maximum number of times the minimum distance can occur among $n$ points in $\mathbb{R}^d$, but to the maximum number of times a regular $k$-simplex of minimum side length can occur among $n$ points in $\mathbb{R}^d$.
Toward the end, I define the $\Delta$-function as follows:
Let $\widehat{v_{1}...v_{k}}$ denote a $k$-simplex with vertices $v_{1},...,v_{k} \in \mathbb{R}^d$. Then, $\Delta_{d}^{k}(n) = \max_{|P|=n} | \{\widehat{v_{1}...v_{k}} | v_{1},...,v_{k} \in P, |v_{i}-v_{j}|=1, i\neq j\}|$ defines the maximum number of occurrences of a regular $k$-simplex of minimum edge length determined by an $n$-element point set in $P \subseteq \mathbb{R}^d$.
So, in terms of this $\Delta$-function, we have $f_{2}^{\text{min}}(n) = \Delta_{2}^{1}(n) = \lfloor 3n - \sqrt{12n -3} \rfloor$. I am now interested in extending Harborth's result to $\Delta_{2}^{2}(n)$ and in doing so I have proved the following corollary of two lemmas I proved (I will not mention the proofs).
Any simplicial $k$-complex $\mathcal{K}$ for which $\Delta_{d}^{k}(n)=|\mathcal{K}|$ is connected and homogeneous, where $k \leq d$ for $P \subseteq \mathbb{R}^d$ with $|P|=n$.
I now want to extend Harborth's result to determining $\Delta_{2}^{2}(n)$, that is, determining the maximum number of regular triangles of side length equal to the minimum distance that can occur among an $n$-element point set in the plane.
The start of my proof looks something like this, I am stuck though and am looking for guidance.
By Corollary 1, we have that $\Delta_{2}^{2}(n)=|\mathcal{K}|$ where $\mathcal{K}$ is a connected homogeneous simplicial $2$-complex. By the necessity of homogeneity, $\partial \mathcal{K}$ is a closed polygon (is it necessarily convex?) with at most $n$-vertices. Let the degree of a vertex in $\mathcal{K}$ denote the number of adjacent vertices in $\mathcal{K}$. Then, let $b$ and $b_{d}$ denote the total number of vertices of $\partial \mathcal{K}$, and the number of those vertices that have degree $d$ in $\mathcal{K}$, respectively. Then, $b = b_{2} + b_{3} + b_{4} + b_{5}$. [From here, I don't know where to go, I want to employ Euler's polyhedron formula somehow. If $\partial \mathcal{K}$ is convex, could I say $\Delta_{2}^{2} = 2 - n + E(n)$, where $E(n)$ is a function which determines the number of edges of $\mathcal{K}$? Any ideas for how to determine E(n)?]
I appreciate any suggestions!