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Claim. Among $n\ge4$ points of $\mathbb R^3$ with diameter one, at most $2n-2$ pairs are at distance one, and this is attained. In the notation of Problem 223, $f_3(n)=2n-2$. This is Vázsonyi's conjecture, which Erdős had recorded in his 1946 note on distances; the library's [[../library/distance_problems/erdos_1946_sets_distances_points/_index|card for that note]] records the conjecture and its link to Borsuk's problem.

Covers. The case $d=3$ of the problem: the exact value $f_3(n)=2n-2$ for every $n\ge4$; the values for $n=2,3$ ($1$ and $3$) are trivial. Nothing is claimed about the plane or about $d\ge4$.

Independent proofs. The same theorem was proved independently and at about the same time by Heppes and by Straszewicz, each of whom has his own claim page in this folder; the three proofs are separate results of record, and the site credits all three.

Acceptance. The paper is refereed: B. Grünbaum, A proof of Vázsonyi's conjecture, Bull. Res. Council Israel Sect. A 6 (1956), 77–78, 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 curator of erdosproblems.com, Thomas Bloom, marks the problem solved and credits the three-dimensional case to this paper among the three.