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OP-9341OpenGeometry and topology

Erdős #1088 · geometry

Let fd(n)f_d(n) be the minimal mm such that any set of mm points in Rd\mathbb{R}^d contains a set of nn points such that any two determined distances are distinct. Estimate fd(n)f_d(n). In particular, is it true that, for f…

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Problem statement

Let fd(n)f_d(n) be the minimal mm such that any set of mm points in Rd\mathbb{R}^d contains a set of nn points such that any two determined distances are distinct. Estimate fd(n)f_d(n). In particular, is it true that, for fixed n≥3n\geq 3,

fd(n)=2o(d)?f_d(n)=2^{o(d)}?

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Background

It is easy to prove that fd(n)≤nOd(1)f_d(n) \leq n^{O_d(1)}. Erdős [Er75f] claimed that he and Straus proved fd(n)≤cndf_d(n)\leq c_n^d for some constant cn>0c_n>0.

When d=1d=1 this is the subject of [530], and f1(n)≍n2f_1(n)\asymp n^2.

When n=3n=3 this is the subject of [503]. Erdős could prove f2(3)=7f_2(3)=7 and Croft [Cr62] proved f3(3)=9f_3(3)=9. The results described at [503] demonstrate that fd(3)=d2/2+O(d)f_d(3)=d^2/2+O(d).

The behaviour of fd(n)f_d(n) for fixed dd as n→∞n\to \infty is the subject of [1208].

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 #1088 link
  2. [2]Upstream Lean 4 statement for Erdős #1088 linkExternal statement reference. Not compiled or verified by this platform; upstream files may contain open proof obligations.
  3. [3]Formal Conjectures — Erdős Problem 1088 link

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