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

- ID: `P3026`
- Reference: `erdos-problem-98`
- Page: https://theoremdb.org/statements/P3026
- Record maturity: Reviewed problem with recorded work

## 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?

### Context

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.

### Problem setup

- **Definition (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 (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 (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.** 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.

### What 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?

## 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. 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](#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 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?

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?

### Background and intake notes

- 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.

### Open directions

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

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-98`, 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 98, maintained status record. Erdős Problems record 98, checked 2026-08-01. Problem 98; status field and linked bibliography https://www.erdosproblems.com/98
   - 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
   - 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 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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 98 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 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.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 98. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/98.lean:L67; theorem erdos_98; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/98.lean#L67
   - 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 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.