[#P3022] Non-uniqueness of point sets minimizing distinct distances

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Problem. 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.

A finite mathematical diagram showing two planar point configurations compared up to similarity. — Two planar point configurations compared up to similarity.

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Definitions and notation

1Context

This is Erdős's Problem 91 from his 1987 paper on combinatorial and metric problems in geometry. Erdős conjectured that for large $n$ there exist at least two (and likely many) non-similar optimal configurations. The cases $n=3$ and $n=5$ are known to have unique minimizers (the equilateral triangle and regular pentagon, respectively), while non-uniqueness has been established for $n=4,6,7,8,9$.

2Problem setup

Definition 1 (For a finite set $A\subset\mathbb{R}^2$, the number of distinct distances). For a finite set $A\subset\mathbb{R}^2$, the number of distinct distances is $d(A)=|\{\|x-y\|:x,y\in A, x\neq y\}|$.

Remark 1. For each positive integer $n$, $m(n)=\min\{d(A):A\subset\mathbb{R}^2, |A|=n\}$.

Definition 2 (A set $A$). A set $A$ is optimal for $n$ if $|A|=n$ and $d(A)=m(n)$.

Definition 3 (Two sets $A,B\subset\mathbb{R}^2$ are similar if there exists a map $f:\mathbb{R}^2\to\mathbb{R}^2$ of the form $f(x)=r\cdot R(x)+t$ where $r>0$, $R$). Two sets $A,B\subset\mathbb{R}^2$ are similar if there exists a map $f:\mathbb{R}^2\to\mathbb{R}^2$ of the form $f(x)=r\cdot R(x)+t$ where $r>0$, $R$ is an orthogonal transformation, and $t\in\mathbb{R}^2$, such that $f(A)=B$.

Remark 2. A positive integer $n$ has a unique minimizer if every two optimal sets for $n$ are similar.

3What counts as a solution

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.

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 91 as open. The unresolved remainder is the full displayed statement. 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.^[1]

1Packet records

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 91 as open. The unresolved remainder is the full displayed statement. 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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 91 as open. The unresolved remainder is the full displayed statement. 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.^[1] Relevance to this problem For erdos problem 91, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 91 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 91: For. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 91 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: 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. Open full record →
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.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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.^[1] Relevance to this problem 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 ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened 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. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 91 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 91 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
The pinned Formal Conjectures declaration was matched by problem number and source locator.
The controlled TheoremDB source corpus was checked for an already published record with the same slug.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemNon-uniqueness of point sets minimizing distinct distances

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Grid points with no three collinear and no four concyclicdiscrete geometry
Largest equilateral-triangle-free subset of the fifteen-row triangular latticediscrete geometry
Most lattice points with all pairwise slopes distinct in a ten by ten griddiscrete geometry

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Plain text

“Non-uniqueness of point sets minimizing distinct distances.” TheoremDB. P3022. Problem statement; statement text SHA-256 54acbdb4cfc1fa72d02aa153da106501784bc17fb247ad8858e185b1068bdfd1. https://theoremdb.org/statement/?ref=P3022

BibTeX

@misc{theoremdb-problem-54acbdb4cfc1fa72d02aa153da106501784bc17fb247ad8858e185b1068bdfd1,
  title = {{Non-uniqueness of point sets minimizing distinct distances}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 54acbdb4cfc1fa72d02aa153da106501784bc17fb247ad8858e185b1068bdfd1},
  url = {https://theoremdb.org/statement/?ref=P3022}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

1References

Packet source. Erdős Problems database, Problem 91, maintained status record. Erdős Problems record 91, checked 2026-08-01. Problem 91; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 91; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Non-uniqueness of point sets minimizing distinct distances: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 91 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For Non-uniqueness of point sets minimizing distinct distances: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 91. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/91.lean:L147; theorem erdos_91; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For Non-uniqueness of point sets minimizing distinct distances: Supplies the pinned formal declaration whose human-readable editorial statement is published here.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

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[#P3022] Non-uniqueness of point sets minimizing distinct distances

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