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OP-9337OpenGeometry and topology

Erdős #1084 · geometry / distances

Let fd(n)f_d(n) be minimal such that in any collection of nn points in Rd\mathbb{R}^d, all of distance at least 11 apart, there are at most fd(n)f_d(n) many pairs of points which are distance 11 apart. Estimate fd(n)f_d(n).

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Problem statement

Let fd(n)f_d(n) be minimal such that in any collection of nn points in Rd\mathbb{R}^d, all of distance at least 11 apart, there are at most fd(n)f_d(n) many pairs of points which are distance 11 apart. Estimate fd(n)f_d(n).

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Background

This is sometimes known as the contact number problem.

It is easy to see that f1(n)=n−1f_1(n)=n-1 and f2(n)<3nf_2(n)<3n (since there can be at most 66 points of distance 11 from any point). Erdős [Er46b] showed

f2(n)<3n−cn1/2f_2(n)<3n-cn^{1/2}

for some constant c>0c>0, which the triangular lattice shows is the best possible up to the value of cc. In [Er75f] he speculated that the triangular lattice is exactly the best possible, and in particular

f2(3n2+3n+1)=9n2+3n.f_2(3n^2+3n+1)=9n^2+3n.

Harborth [Ha74b] proved this, and more generally

f2(n)=⌊3n−12n−3⌋f_2(n)=\lfloor 3n-\sqrt{12n-3}\rfloor

for all n≥2n\geq 2.

In [Er75f] he claims the existence of c1,c2>0c_1,c_2>0 such that

6n−c1n2/3<f3(n)<6n−c2n2/3.6n-c_1n^{2/3}< f_3(n) < 6n-c_2n^{2/3}.

An upper bound of

f3(n)<6n−0.926n2/3f_3(n) < 6n-0.926n^{2/3}

for all n≥2n\geq 2 was proved by Bezdek and Reid [BeRe13].

In general, it is known that

(d−o(1))n≤fd(n)≤2O(d)n,(d-o(1))n \leq f_d(n) \leq 2^{O(d)}n,

the lower bound coming from points arranged in an integer grid and the upper bound from the fact that 2O(d)2^{O(d)} many non-intersecting congruent balls can touch a fixed ball (the kissing number problem).

A recent survey on contact numbers for sphere packings is by Bezdek and Khan [BeKh18].

See [223] for the analogous problem with maximal distance 11.

Progress & source assessment

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.

References
  1. [1]Thomas Bloom (ed.), Erdős Problems #1084 link
  2. [2]Upstream Lean 4 statement for Erdős #1084 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 1084 link

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