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OP-9276OpenGeometry and topology

Erdős #956 · geometry / distances

If C,D⊆R2C,D\subseteq \mathbb{R}^2 then the distance between CC and DD is defined by

δ(C,D)=inf⁡c∈Cd∈D∥c−d∥.\delta(C,D)=\inf_{\substack{c\in C\\ d\in D}}\| c-d\|.

Let h(n)h(n) be the maximal number of unit distances between disjoint convex…

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Problem statement

If C,D⊆R2C,D\subseteq \mathbb{R}^2 then the distance between CC and DD is defined by

δ(C,D)=inf⁡c∈Cd∈D∥c−d∥.\delta(C,D)=\inf_{\substack{c\in C\\ d\in D}}\| c-d\|.

Let h(n)h(n) be the maximal number of unit distances between disjoint convex translates. That is, the maximal mm such that there is a compact convex set C⊂R2C\subset \mathbb{R}^2 and a set XX of size nn such that all (C+x)x∈X(C+x)_{x\in X} are disjoint and there are mm pairs x1,x2∈Xx_1,x_2\in X such that

δ(C+x1,C+x2)=1.\delta(C+x_1,C+x_2)=1.

Determine h(n)h(n) - in particular, prove that there exists a constant c>0c>0 such that h(n)>n1+ch(n)>n^{1+c} for all large nn.

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Background

A problem of Erdős and Pach [ErPa90], who proved that h(n)≪n4/3h(n) \ll n^{4/3}. They also consider the related function where we consider nn disjoint convex sets (not necessarily translates), for which they give an upper bound of ≪n7/5\ll n^{7/5}.

It is trivial that h(n)≥f(n)h(n)\geq f(n), where f(n)f(n) is the maximal number of unit distances determined by nn points in R2\mathbb{R}^2 (see [90]).

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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 #956 link

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