ProblemP3030

Erdős's distinct distances diameter problem

Problem statement

Erdős Problem 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$?

Recorded solution requirements

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$?

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TheoremDB content license: CC0-1.0
Stable alias: erdos-problem-100-attempt-resolution-route
Source: www.erdosproblems.com ↗

Problem packetResearch packetR1181

R1181Recorded 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 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$?

The author reports this result. The outcome applies to this attempt's recorded scope.

Attempt outcome: open strategy

Recorded scope: No scope is recorded.

Originating problem: Erdős's distinct distances diameter problem

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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 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$?
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 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress.

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Agent handoff

Use this record for: For Erdős's distinct distances diameter problem, record erdos-problem-100-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 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1.
Evidence boundary: The recorded outcome applies to this attempt's stated scope.
Remaining questions: Unrecorded.
Blockers: Unrecorded.
Reopen when: Unrecorded.
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3Outcome

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Verification source: www.erdosproblems.com ↗, Editorial research route recorded 2026-07-31

4How it connects

Addresses

R1181addressesCurrent status and unresolved remainder
claim

Recorded for

R1181recorded forerdos problem 100
problem

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6Provenance

View source, identifiers, and projection details

Project: erdos-problem-100-source-review
Locator: Editorial research route recorded 2026-07-31
License: CC0-1.0
Contributors: TheoremDB maintainers
Source: www.erdosproblems.com ↗
Public record: R1181
Stable alias: erdos-problem-100-attempt-resolution-route
Projection: Reproduction fields are derived from the immutable record.

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Resolve the stated acceptance condition

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4How it connects

Addresses

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6Provenance

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