Claims

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1934_01_01_hopf_pannwitz: Among $n$ points of the plane with diameter one, the distance one occurs at most $n$ times, and $n$ is attained for $n\ge3$, so $f_2(n)=n$.

1956_01_01_grunbaum: Among $n\ge4$ points of three-dimensional space with diameter one, the distance one occurs at most $2n-2$ times, so $f_3(n)=2n-2$ for $n\ge4$.

1956_09_01_heppes: Among $n\ge4$ points of three-dimensional space with diameter one, the distance one occurs at most $2n-2$ times, so $f_3(n)=2n-2$ for $n\ge4$.

1957_01_01_straszewicz: Among $n\ge4$ points of three-dimensional space with diameter one, the distance one occurs at most $2n-2$ times, so $f_3(n)=2n-2$ for $n\ge4$.

1960_01_01_erdos: For $d\ge4$ and $p=\lfloor d/2\rfloor$, the maximum number of diameters among $n$ points of diameter one is $(\frac{p-1}{2p}+o(1))n^2$.

2007_07_02_swanepoel: For $d\ge4$ and all $n$ large in terms of $d$, the $n$-point sets of diameter one with the most pairs at distance one are Lenz configurations, which gives the exact value of $f_d(n)$.

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