ProblemP3022

Non-uniqueness of point sets minimizing distinct distances

Problem statement

Erdős Problem 91: For a finite set $A\subset\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\subset\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is called optimal (for $n$). Two finite sets $A,B\subset\mathbb{R}^2$ are called similar if there exists a dilation of the plane mapping $A$ onto $B$ (where a dilation is a composition of a scaling and an isometry). For each $n$, we say that $n$ has a unique minimizer if all optimal sets for $n$ are similar to one another. Determine whether, for all sufficiently large integers $n$, the number $n$ does not have a unique minimizer.

Recorded solution requirements

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 91: For a finite set $A\subset\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\subset\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is called optimal (for $n$). Two finite sets $A,B\subset\mathbb{R}^2$ are called similar if there exists a dilation of the plane mapping $A$ onto $B$ (where a dilation is a composition of a scaling and an isometry). For each $n$, we say that $n$ has a unique minimizer if all optimal sets for $n$ are similar to one another. Determine whether, for all sufficiently large integers $n$, the number $n$ does not have a unique minimizer.

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TheoremDB content license: CC0-1.0
Stable alias: erdos-problem-91-attempt-resolution-route
Source: www.erdosproblems.com ↗

Problem packetResearch packetR1269

R1269Recorded attemptOutcome: Open

Resolve the stated acceptance condition

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Authored summary

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 91: For a finite set $A\subset\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\subset\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is called optimal (for $n$). Two finite sets $A,B\subset\mathbb{R}^2$ are called similar if there exists a dilation of the plane mapping $A$ onto $B$ (where a dilation is a composition of a scaling and an isometry). For each $n$, we say that $n$ has a unique minimizer if all optimal sets for $n$ are similar to one another. Determine whether, for all sufficiently large integers $n$, the number $n$ does not have a unique minimizer.

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Attempt outcome: open strategy

Recorded scope: No scope is recorded.

Originating problem: Non-uniqueness of point sets minimizing distinct distances

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Authored record and scope

Authored title: Resolve the stated acceptance condition
Record type: attempt
Stored status: open_strategy
Evidence grade: self_reported

Work and source credit

Recorded action: No action description supplied.
Authored result summary: Read complete authored result summary
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 91: For a finite set $A\subset\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\subset\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is called optimal (for $n$). Two finite sets $A,B\subset\mathbb{R}^2$ are called similar if there exists a dilation of the plane mapping $A$ onto $B$ (where a dilation is a composition of a scaling and an isometry). For each $n$, we say that $n$ has a unique minimizer if all optimal sets for $n$ are similar to one another. Determine whether, for all sufficiently large integers $n$, the number $n$ does not have a unique minimizer.
Reported outcome: No separate outcome supplied.
Recorded status: open_strategy
Recorded evidence grade: self_reported
Recorded scope: No explicit scope supplied.

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2Authored explanation

Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 91: For a finite set $A\subset\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$. For each positive integer $n$, define $m(n)$ to be the minimum value of $d(A)$ over all sets $A\subset\mathbb{R}^2$ with $|A|=n$. A set $A$ with $|A|=n$ and $d(A)=m(n)$ is called optimal (for $n$). Two finite sets $A,B\subset\mathbb{R}^2$ are called similar if there exists a dilation of the plane mapping $A$ onto $B$ (where a dilation is a composition of a scaling and an isometry). For each $n$, we say that $n$ has a unique minimizer if all optimal sets for $n$ are similar to one another. Determine whether, for all sufficiently large integers $n$, the number $n$ does not have a unique minimizer. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress.

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Use this record for: For Non-uniqueness of point sets minimizing distinct distances, record erdos-problem-91-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 91: For a finite set $A\subset\mathbb{R}^2$, let $d(A)$ denote the number of distinct Euclidean distances between pairs of distinct points of $A$.
Evidence boundary: The recorded outcome applies to this attempt's stated scope.
Remaining questions: Unrecorded.
Blockers: Unrecorded.
Reopen when: Unrecorded.
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Verification source: www.erdosproblems.com ↗, Editorial research route recorded 2026-07-31

4How it connects

Addresses

R1269addressesCurrent status and unresolved remainder
claim

Recorded for

R1269recorded forerdos problem 91
problem

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6Provenance

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Project: erdos-problem-91-source-review
Locator: Editorial research route recorded 2026-07-31
License: CC0-1.0
Contributors: TheoremDB maintainers
Source: www.erdosproblems.com ↗
Public record: R1269
Stable alias: erdos-problem-91-attempt-resolution-route
Projection: Reproduction fields are derived from the immutable record.

A route someone took, recorded so the next person can reuse it or avoid it.

Resolve the stated acceptance condition

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