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Statement

Setting (pp. 1, 4). $c(n,3)$ is the largest number of touching pairs among $n$ non-overlapping unit balls in $\mathbb E^3$. $c_{\mathrm{fcc}}(n)$ is the largest contact number of a packing of $n$ unit balls whose centers are all lattice points of the face-centered cubic lattice with shortest non-zero lattice vector of length $2$.

Theorem 4.1 (p. 5).

(i) $c(n,3)<6n-0.926\,n^{2/3}$ for all $n\ge2$.

(ii) $c_{\mathrm{fcc}}(n)<6n-\frac{3\sqrt[3]{18\pi}}{\pi}\,n^{2/3}=6n-3.665\ldots n^{2/3}$ for all $n\ge2$.

(iii) $6n-\sqrt[3]{486}\,n^{2/3}<2k(2k^2-3k+1)\le c_{\mathrm{fcc}}(n)\le c(n,3)$ for all $n=\frac{k(2k^2+1)}{3}$ with $k\ge2$.

The survey attributes (i) to Bezdek and Reid (its reference [12], J. Geom. 104 (2013)), proved with the method of Bezdek (its reference [11], Discrete Comput. Geom. 48 (2012)), and (ii) and (iii) to that 2012 paper. It derives (2) (p. 5): $0.926<(6n-c(n,3))/n^{2/3}<\sqrt[3]{486}=7.862\ldots$ for all $n=k(2k^2+1)/3$ with $k\ge2$.

Proof pointer

Not proved in the survey; the proofs are in the cited papers.

Read depth

Claims checked: the definitions and the three parts were read on the page images of the print, with (2). The cited proofs were not read here. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs: the cited papers of Bezdek (2012) and Bezdek and Reid (2013); the latter has its own source card.

Source. K. Bezdek and M. A. Khan, Contact numbers for sphere packings, in New Trends in Intuitive Geometry, Bolyai Society Mathematical Studies, Springer (2018), 25--47, doi:10.1007/978-3-662-57413-3_2; the label and page are those of arXiv:1601.00145v2, the edition read, named on the source card.

Bears on

• Problem 1084: after scaling by $1/2$, $c(n,3)$ is $f_3(n)$. Part (i) restates the upper bound $f_3(n)<6n-0.926\,n^{2/3}$ for $n\ge2$, credited on the problem's claim page to Bezdek and Reid; part (iii) gives $f_3(n)>6n-\sqrt[3]{486}\,n^{2/3}$ only for the octahedral numbers $n=k(2k^2+1)/3$, $k\ge2$. On those $n$ the two bounds have the order $6n-\Theta(n^{2/3})$ of Erdős's estimate for $d=3$; the survey does not give the lower bound for other $n$.