Problem 1084

Let $f_d(n)$ be minimal such that in any collection of $n$ points in $\mathbb{R}^d$ , all of distance at least $1$ apart, there are at most $f_d(n)$ many pairs of points which are distance $1$ apart. Estimate $f_d(n)$ .

Source: erdosproblems.com/1084

Geometry
Distances

Wiki page

Status

Open

No claim settles this problem.

Open, in the site's label (OPEN; page last edited 8 February 2026). The site's remarks credit exact values for $d=1$ and $d=2$ and an upper bound for $d=3$ , recorded as partial claims in claims/; the problem asks for $f_d(n)$ in every dimension, so the standing in the frontmatter, derived from the claim pages, is open.

BeKh18: Bezdek, Károly and Khan, Muhammad A., Contact numbers for sphere packings. (2018), 25-47.
BeRe13: Bezdek, Károly and Reid, Samuel, Contact graphs of unit sphere packings revisited. J. Geom. (2013), 57-83.
Er46b: Erdős, P., On sets of distances of $n$ points. Amer. Math. Monthly (1946), 248-250.
Er75f: Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.
Ha74b: Harborth, Heiko, Lösung zu Problem 664A. Elem. Math. (1974), 14-15.

Problem 1084

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Proof claims

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