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Claim. Among $n\ge2$ points of the plane with diameter one, at most $n$ pairs are at distance one, and for every $n\ge3$ some $n$-point set of diameter one has exactly $n$ such pairs. In the notation of Problem 223, $f_2(n)=n$ for $n\ge3$, while two points give the single pair $f_2(2)=1$.

Covers. The case $d=2$ of the problem: the exact value of $f_2(n)$ for every $n\ge2$, namely $n$ for $n\ge3$ and $1$ for $n=2$. Nothing is claimed about $d\ge3$.

The argument. The result is Aufgabe 167 of the Jahresbericht der Deutschen Mathematiker-Vereinigung 43 (1934), 114, a posed problem whose published solutions followed, cited with its volume as the bibliography of [[../library/distance_problems/swanepoel_2009_unit_distances_diameters_euclidean_spaces/_index|Swanepoel's 2009 paper]] gives it; the site records that Erdős's 1946 note describes a short proof. That note's Theorem 3, that the maximum distance among $n$ planar points occurs at most $n$ times, is summarized on the library's [[../library/distance_problems/erdos_1946_sets_distances_points/_index|card for Erdős's note]]; the bound is attained, for odd $n$, by the vertices of a regular $n$-gon, whose longest diagonals form a cycle of length $n$. Erdős also records there Vázsonyi's conjecture for the three-dimensional case.

Acceptance. The curator of erdosproblems.com, Thomas Bloom, marks the problem solved and credits the planar case to Hopf and Pannwitz, with Erdős's 1946 Monthly note (Amer. Math. Monthly 53 (1946), 248–250) as the published exposition of the proof. The 1934 item itself is a problem posting, so no refereed publication of the claimants' own proof is listed.