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Claim. For every $n\ge5$,

$$f_4(n)= \begin{cases} \lfloor n^2/4\rfloor+n & \text{if } 8\mid n \text{ or } 10\mid n,\\ \lfloor n^2/4\rfloor+n-1 & \text{otherwise}, \end{cases}$$

in the notation of Problem 1085: the exact value of $f_4(n)$ for every $n\ge5$. The paper is P. Brass, On the maximum number of unit distances among $n$ points in dimension four, in: Intuitive Geometry (Budapest, 1995), Bolyai Society Mathematical Studies 6, János Bolyai Mathematical Society, Budapest, 1997, 277–290. Brass's determination relies on a number-theoretic statement completed by P. van Wamelen, The maximum number of unit distances among $n$ points in dimension four, Beiträge zur Algebra und Geometrie 40 (1999), 475–477; the formula above is the combined result as the introduction of Swanepoel's paper [Sw09] states it.

Covers. The exact value of $f_4(n)$ for every $n\ge5$, which sharpens the additive-constant estimate of Erdős's 1967 paper in the case $d=4$. Nothing is claimed for other dimensions.

Depends on. No page of this wiki.

Acceptance. None documented. The result appeared in a proceedings volume of the Bolyai Society, for which no evidence of refereeing is recorded, so refereed is not listed; no outside review is recorded; and the site's remarks credit Brass with determining $f_4(n)$ exactly while labeling the problem OPEN, which is not an acceptance of the problem or of a part. The statement is taken from the site's remarks and from the introduction of [Sw09], which cites both papers and the Mathematical Reviews entry MR 98j:52030 of Brass's; neither paper has a library card. The claim is claimed.