[#P3028] Equilateral Triangles in Diameter-Minimizing Point Sets with Unit Minimum Distance

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Problem. Erdős Problem 99: For a finite set $A$ of points in the plane, we say that $A$ has minimum distance $1$ if every pair of distinct points in $A$ is at distance at least $1$, and there exists at least one pair of points in $A$ at distance exactly $1$. The diameter of $A$ is the maximum distance between any two points of $A$. For each positive integer $n$, consider the family of all $n$-point sets in the plane with minimum distance $1$, and let $A$ be a set in this family that achieves the minimum possible diameter. Does it follow that for all sufficiently large $n$, such a diameter-minimizing set $A$ must contain three points that form an equilateral triangle of side length $1$?

A finite mathematical diagram showing a planar point set with an equilateral triangle marked. — A planar point set with an equilateral triangle marked.

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Definitions and notation

1Context

This problem belongs to discrete geometry, concerning extremal configurations of point sets with separation constraints. It asks whether optimal configurations for minimizing diameter under a unit minimum distance constraint must contain a specific rigid substructure (an equilateral triangle of side length 1) when the number of points is large enough.

2Problem setup

Remark 1. A set $A$ of points in the plane has minimum distance $1$ if every pair of distinct points $p, q \in A$ satisfies $\operatorname{dist}(p,q) \geq 1$, and there exist points $p, q \in A$ with $\operatorname{dist}(p,q) = 1$.

Remark 2. Three points $p, q, r$ in the plane form an equilateral triangle of side length $1$ if $\operatorname{dist}(p,q) = \operatorname{dist}(q,r) = \operatorname{dist}(p,r) = 1$.

Definition 1 (For a set $A$ in a family $\mathcal{F}$ of sets, $A$). For a set $A$ in a family $\mathcal{F}$ of sets, $A$ is minimal for a real-valued function $f$ on $\mathcal{F}$ if $f(A) \leq f(B)$ for all $B \in \mathcal{F}$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 99: For a finite set $A$ of points in the plane, we say that $A$ has minimum distance $1$ if every pair of distinct points in $A$ is at distance at least $1$, and there exists at least one pair of points in $A$ at distance exactly $1$. The diameter of $A$ is the maximum distance between any two points of $A$. For each positive integer $n$, consider the family of all $n$-point sets in the plane with minimum distance $1$, and let $A$ be a set in this family that achieves the minimum possible diameter. Does it follow that for all sufficiently large $n$, such a diameter-minimizing set $A$ must contain three points that form an equilateral triangle of side length $1$?

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 99 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 99: For a finite set $A$ of points in the plane, we say that $A$ has minimum distance $1$ if every pair of distinct points in $A$ is at distance at least $1$, and there exists at least one pair of points in $A$ at distance exactly $1$. The diameter of $A$ is the maximum distance between any two points of $A$. For each positive integer $n$, consider the family of all $n$-point sets in the plane with minimum distance $1$, and let $A$ be a set in this family that achieves the minimum possible diameter. Does it follow that for all sufficiently large $n$, such a diameter-minimizing set $A$ must contain three points that form an equilateral triangle of side length $1$?^[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 99 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 99: For a finite set $A$ of points in the plane, we say that $A$ has minimum distance $1$ if every pair of distinct points in $A$ is at distance at least $1$, and there exists at least one pair of points in $A$ at distance exactly $1$. The diameter of $A$ is the maximum distance between any two points of $A$. For each positive integer $n$, consider the family of all $n$-point sets in the plane with minimum distance $1$, and let $A$ be a set in this family that achieves the minimum possible diameter. Does it follow that for all sufficiently large $n$, such a diameter-minimizing set $A$ must contain three points that form an equilateral triangle of side length $1$?	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 99 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 99: For a finite set $A$ of points in the plane, we say that $A$ has minimum distance $1$ if every pair of distinct points in $A$ is at distance at least $1$, and there exists at least one pair of points in $A$ at distance exactly $1$. The diameter of $A$ is the maximum distance between any two points of $A$. For each positive integer $n$, consider the family of all $n$-point sets in the plane with minimum distance $1$, and let $A$ be a set in this family that achieves the minimum possible diameter. Does it follow that for all sufficiently large $n$, such a diameter-minimizing set $A$ must contain three points that form an equilateral triangle of side length $1$?^[1] Relevance to this problem For erdos problem 99, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 99 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 99: 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 99 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 99: For a finite set $A$ of points in the plane, we say that $A$ has minimum distance $1$ if every pair of distinct points in $A$ is at distance at least $1$, and there exists at least one pair of points in $A$ at distance exactly $1$. The diameter of $A$ is the maximum distance between any two points of $A$. For each positive integer $n$, consider the family of all $n$-point sets in the plane with minimum distance $1$, and let $A$ be a set in this family that achieves the minimum possible diameter. Does it follow that for all sufficiently large $n$, such a diameter-minimizing set $A$ must contain three points that form an equilateral triangle of side length $1$? Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 99: For a finite set $A$ of points in the plane, we say that $A$ has minimum distance $1$ if every pair of distinct points in $A$ is at distance at least $1$, and there exists at least one pair of points in $A$ at distance exactly $1$. The diameter of $A$ is the maximum distance between any two points of $A$. For each positive integer $n$, consider the family of all $n$-point sets in the plane with minimum distance $1$, and let $A$ be a set in this family that achieves the minimum possible diameter. Does it follow that for all sufficiently large $n$, such a diameter-minimizing set $A$ must contain three points that form an equilateral triangle of side length $1$?	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 99: For a finite set $A$ of points in the plane, we say that $A$ has minimum distance $1$ if every pair of distinct points in $A$ is at distance at least $1$, and there exists at least one pair of points in $A$ at distance exactly $1$. The diameter of $A$ is the maximum distance between any two points of $A$. For each positive integer $n$, consider the family of all $n$-point sets in the plane with minimum distance $1$, and let $A$ be a set in this family that achieves the minimum possible diameter. Does it follow that for all sufficiently large $n$, such a diameter-minimizing set $A$ must contain three points that form an equilateral triangle of side length $1$?^[1] Relevance to this problem For Equilateral Triangles in Diameter-Minimizing Point Sets with Unit Minimum Distance, record erdos-problem-99-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 99: For a finite set $A$ of points in the plane, we say that $A$ has minimum distance $1$ if every pair of distinct points in $A$ is at distance at least $1$, and there exists at least one pair of points in $A$ at distance exactly $1$. 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 99: For a finite set $A$ of points in the plane, we say that $A$ has minimum distance $1$ if every pair of distinct points in $A$ is at distance at least $1$, and there exists at least one pair of points in $A$ at distance exactly $1$. The diameter of $A$ is the maximum distance between any two points of $A$. For each positive integer $n$, consider the family of all $n$-point sets in the plane with minimum distance $1$, and let $A$ be a set in this family that achieves the minimum possible diameter. Does it follow that for all sufficiently large $n$, such a diameter-minimizing set $A$ must contain three points that form an equilateral triangle of side length $1$? 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 99 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 99 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.

ProblemEquilateral Triangles in Diameter-Minimizing Point Sets with Unit Minimum Distance

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

“Equilateral Triangles in Diameter-Minimizing Point Sets with Unit Minimum Distance.” TheoremDB. P3028. Problem statement; statement text SHA-256 5d2e794dfb2db0907f158b419647034e183dd679a81bb505703a4659878cffdf. https://theoremdb.org/statement/?ref=P3028

BibTeX

@misc{theoremdb-problem-5d2e794dfb2db0907f158b419647034e183dd679a81bb505703a4659878cffdf,
  title = {{Equilateral Triangles in Diameter-Minimizing Point Sets with Unit Minimum Distance}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 5d2e794dfb2db0907f158b419647034e183dd679a81bb505703a4659878cffdf},
  url = {https://theoremdb.org/statement/?ref=P3028}
}

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 99, maintained status record. Erdős Problems record 99, checked 2026-08-01. Problem 99; 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 99; 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 Equilateral Triangles in Diameter-Minimizing Point Sets with Unit Minimum Distance: 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 99 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 Equilateral Triangles in Diameter-Minimizing Point Sets with Unit Minimum Distance: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 99. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/99.lean:L44; theorem erdos_99; 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 Equilateral Triangles in Diameter-Minimizing Point Sets with Unit Minimum Distance: 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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[#P3028] Equilateral Triangles in Diameter-Minimizing Point Sets with Unit Minimum Distance

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