[#P3026] Asymptotic growth of minimum distinct distances for point sets in general position

A finite mathematical diagram showing planar points in general position with distance classes marked. — Planar points in general position with distance classes marked.

Problem. 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?

1Context

This problem originates from Paul Erdős's investigations in combinatorial geometry concerning the relationship between the number of points in a planar set and the number of distinct distances they determine, under the additional restriction of general position.

2Problem setup

Definition 1 (A set of points in $\mathbb{R}^2$). A set of points in $\mathbb{R}^2$ is in general position if no three points lie on a common line and no four points lie on a common circle.

Definition 2 (For a finite set of points in $\mathbb{R}^2$, the distinct distances of the set). For a finite set of points in $\mathbb{R}^2$, the distinct distances of the set is the number of different Euclidean distances realized between pairs of points in the set.

Definition 3 (For each natural number $n$, the function $h(n)$). For each natural number $n$, the function $h(n)$ is defined as the infimum of the distinct distances over all $n$-point subsets of $\mathbb{R}^2$ in general position.

Remark 1. This problem originates from Paul Erdős's investigations in combinatorial geometry concerning the relationship between the number of points in a planar set and the number of distinct distances they determine, under the additional restriction of general position.

3What counts as a solution

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?

1Status

Current status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 98 as open. The unresolved remainder is the full displayed statement. 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?^[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 98 as open. The unresolved remainder is the full displayed statement. 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?^[1] Relevance to this problem For erdos problem 98, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 98 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 98: 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 98 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 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? Open full record →
	Trace	Reported
attempt · Route 1 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?^[1] Relevance to this problem 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 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 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. Related: Claim 1 Open full record →
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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 98 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 98 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

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

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, “Asymptotic growth of minimum distinct distances for point sets in general position,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-98

This page as plain text: erdos-problem-98.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 98, maintained status record. Erdős Problems record 98, checked 2026-08-01. Problem 98; 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 98; 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 Asymptotic growth of minimum distinct distances for point sets in general position: 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 98 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 Asymptotic growth of minimum distinct distances for point sets in general position: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 98. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/98.lean:L67; theorem erdos_98; 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 Asymptotic growth of minimum distinct distances for point sets in general position: 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.

[#P3026] Asymptotic growth of minimum distinct distances for point sets in general position

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

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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