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

- ID: `P3028`
- Reference: `erdos-problem-99`
- Page: https://theoremdb.org/statements/P3028
- Record maturity: Reviewed problem with recorded work

## 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 $1$?

### Context

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.

### Problem setup

- **Definition (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 (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 (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.** 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.

### What 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 $1$?

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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 $1$?

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 $1$?

### Background and intake notes

- 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-99`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Erdős Problems database, Problem 99, maintained status record. Erdős Problems record 99, checked 2026-08-01. Problem 99; status field and linked bibliography https://www.erdosproblems.com/99
   - 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
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Records the current open status and links the literature attached to this exact numbered problem.
   - 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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 99 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source 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.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 99. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/99.lean:L44; theorem erdos_99; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/99.lean#L44
   - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31
   - Source 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.