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

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

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 $1$?

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

Definition 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 $\operatorname{dist}(p,q) = 1$). 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 $\operatorname{dist}(p,q) = 1$.

Definition 2 (Three points $p, q, r$ in the plane form an equilateral triangle of side length $1$ if $\operatorname{dist}(p,q) = \operatorname{dist}(q,r) = \operatorname{dist}(p,r) = 1$). 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 3 (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 $B \in \mathcal{F}$.

Remark 1. 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.

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

Current 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

Record	Kind	Assessment
	Result	Supported
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 →
	Trace	Reported
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 →
No records match these filters.

Notes and companion materialContext, examples, and computations

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 connectTyped relations and evidence flow

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

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

Conway’s thrackle conjecturediscrete geometry
Borsuk’s conjecture in four dimensionsdiscrete geometry
Completing a line arrangement to triangular bounded cellsdiscrete geometry

How to cite

TheoremDB contributors, “Equilateral Triangles in Diameter-Minimizing Point Sets with Unit Minimum Distance,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-99

This page as plain text: erdos-problem-99.md

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.

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

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Report a problem

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance