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Claim. Theorem 1(i) of Bezdek and Reid states that the number of touching pairs in any packing of $n\ge2$ unit balls in $\mathbb{R}^3$ is less than $6n-0.926\,n^{2/3}$. The centers of a packing of balls of diameter $1$ are exactly the sets of points at mutual distance at least $1$, and touching pairs are the pairs at distance exactly $1$; after scaling, the theorem reads

$$f_3(n)<6n-0.926\,n^{2/3}\qquad\text{for every }n\ge2$$

in the notation of Problem 1084. The paper's card records the theorem; its proof uses truncated Voronoi cells, a density estimate for unions of balls, an isoperimetric inequality and bounds for packings of spherical caps.

Covers. The upper half of Erdős's claim in [Er75f] that $6n-c_1n^{2/3}<f_3(n)<6n-c_2n^{2/3}$ for constants $c_1,c_2>0$: the upper inequality holds with $c_2=0.926$ for every $n\ge2$. The lower half is not claimed; the paper recalls packings with more than $6n-7.862\,n^{2/3}$ touching pairs only for $n=(2k^3+k)/3$. Nothing exact is claimed for any $d\ge3$.

Depends on. No page of this wiki.

Acceptance. Refereed: K. Bezdek and S. Reid, Contact graphs of unit sphere packings revisited, Journal of Geometry 104 (2013), no. 1, 57–83; the preprint is arXiv:1210.5756. The site's remarks credit the bound to this paper, but the site labels the problem OPEN, so the remark is not reviewed evidence. The page is dated to the issue, April 2013.