ProblemP3026

Asymptotic growth of minimum distinct distances for point sets in general position

Problem statement

Erdős Problem 98: For each natural number $n$, let $h(n)$ denote the minimum number of distinct distances determined by any set of $n$ points in the Euclidean plane $\mathbb{R}^2$ that is in general position, meaning no three points are collinear and no four points are concyclic (lie on a common circle). Does the ratio $h(n)/n$ tend to infinity as $n \to \infty$? That is, does \[ \lim_{n \to \infty} \frac{h(n)}{n} = \infty \] hold?

Recorded solution requirements

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 98: For each natural number $n$, let $h(n)$ denote the minimum number of distinct distances determined by any set of $n$ points in the Euclidean plane $\mathbb{R}^2$ that is in general position, meaning no three points are collinear and no four points are concyclic (lie on a common circle). Does the ratio $h(n)/n$ tend to infinity as $n \to \infty$? That is, does \[ \lim_{n \to \infty} \frac{h(n)}{n} = \infty \] hold?

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Record facts

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TheoremDB content license: CC0-1.0
Stable alias: erdos-problem-98-attempt-resolution-route
Source: www.erdosproblems.com ↗

Problem packetResearch packetR1273

R1273Recorded 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 98: For each natural number $n$, let $h(n)$ denote the minimum number of distinct distances determined by any set of $n$ points in the Euclidean plane $\mathbb{R}^2$ that is in general position, meaning no three points are collinear and no four points are concyclic (lie on a common circle). Does the ratio $h(n)/n$ tend to infinity as $n \to \infty$? That is, does \[ \lim_{n \to \infty} \frac{h(n)}{n} = \infty \] hold?

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: Asymptotic growth of minimum distinct distances for point sets in general position

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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 98: For each natural number $n$, let $h(n)$ denote the minimum number of distinct distances determined by any set of $n$ points in the Euclidean plane $\mathbb{R}^2$ that is in general position, meaning no three points are collinear and no four points are concyclic (lie on a common circle). Does the ratio $h(n)/n$ tend to infinity as $n \to \infty$? That is, does \[ \lim_{n \to \infty} \frac{h(n)}{n} = \infty \] hold?
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 98: For each natural number $n$, let $h(n)$ denote the minimum number of distinct distances determined by any set of $n$ points in the Euclidean plane $\mathbb{R}^2$ that is in general position, meaning no three points are collinear and no four points are concyclic (lie on a common circle). Does the ratio $h(n)/n$ tend to infinity as $n \to \infty$? That is, does \[ \lim_{n \to \infty} \frac{h(n)}{n} = \infty \] hold? 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 Asymptotic growth of minimum distinct distances for point sets in general position, record erdos-problem-98-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 98: For each natural number $n$, let $h(n)$ denote the minimum number of distinct distances determined by any set of $n$ points in the Euclidean plane $\mathbb{R}^2$ that is in general position, meaning no three points are collinear and no four points are concyclic (lie on a common circle).
Evidence boundary: The recorded outcome applies to this attempt's stated scope.
Remaining questions: Unrecorded.
Blockers: Unrecorded.
Reopen when: Unrecorded.
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Replay material: source only

3Outcome

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

4How it connects

Addresses

R1273addressesCurrent status and unresolved remainder
claim

Recorded for

R1273recorded forerdos problem 98
problem

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6Provenance

View source, identifiers, and projection details

Project: erdos-problem-98-source-review
Locator: Editorial research route recorded 2026-07-31
License: CC0-1.0
Contributors: TheoremDB maintainers
Source: www.erdosproblems.com ↗
Public record: R1273
Stable alias: erdos-problem-98-attempt-resolution-route
Projection: Reproduction fields are derived from the immutable record.

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Resolve the stated acceptance condition

Work and source credit

2Authored explanation

3Outcome

4How it connects

Addresses

Recorded for

6Provenance

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