# P3020: Erdős Distinct Distances Lower Bound Conjecture

- ID: `P3020`
- Reference: `erdos-problem-89`
- Page: https://theoremdb.org/statements/P3020
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 89: For a positive integer $n$, let $f(n)$ denote the minimum, over all sets of $n$ distinct points in the Euclidean plane $\mathbb{R}^2$, of the number of distinct pairwise distances determined by those points. Determine whether there exists a positive constant $C$ such that for all sufficiently large $n$, every set of $n$ distinct points in $\mathbb{R}^2$ determines at least $C \cdot \frac{n}{\sqrt{\log n}}$ distinct distances. Equivalently, determine whether $f(n)$ satisfies $f(n) \gg \frac{n}{\sqrt{\log n}}$ as $n \to \infty$.

### Context

This problem originates from Paul Erdős's 1946 study of distinct distances in planar point sets. The notation $\gg$ denotes that the left-hand side is bounded below by a positive constant multiple of the right-hand side for sufficiently large values. The function $f(n)$ represents the extremal quantity: the worst-case (minimum) number of distinct distances that any $n$-point planar configuration must realize.

### Problem setup

- **Definition (For a finite set $P \subset \mathbb{R}^2$, the number of distinct distances determined by $P$).** For a finite set $P \subset \mathbb{R}^2$, the number of distinct distances determined by $P$ is the cardinality of the set $\{|p - q| : p, q \in P, p \neq q\}$, where $|p - q|$ denotes the Euclidean distance between points $p$ and $q$.
- **Definition (For functions $g, h : \mathbb{N} \to \mathbb{R}_{>0}$, the asymptotic notation $g(n) \gg h(n)$ as $n \to \infty$).** For functions $g, h : \mathbb{N} \to \mathbb{R}_{>0}$, the asymptotic notation $g(n) \gg h(n)$ as $n \to \infty$ means that there exist positive constants $C$ and $N$ such that $g(n) \geq C \cdot h(n)$ for all $n \geq N$.
- **Definition (The natural logarithm).** The natural logarithm is denoted by $\log n$, and $\sqrt{\log n} = (\log n)^{1/2}$.
- **Remark.** This problem originates from Paul Erdős's 1946 study of distinct distances in planar point sets. The notation $\gg$ denotes that the left-hand side is bounded below by a positive constant multiple of the right-hand side for sufficiently large values. The function $f(n)$ represents the extremal quantity: the worst-case (minimum) number of distinct distances that any $n$-point planar configuration must realize.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 89: For a positive integer $n$, let $f(n)$ denote the minimum, over all sets of $n$ distinct points in the Euclidean plane $\mathbb{R}^2$, of the number of distinct pairwise distances determined by those points. Determine whether there exists a positive constant $C$ such that for all sufficiently large $n$, every set of $n$ distinct points in $\mathbb{R}^2$ determines at least $C \cdot \frac{n}{\sqrt{\log n}}$ distinct distances. Equivalently, determine whether $f(n)$ satisfies $f(n) \gg \frac{n}{\sqrt{\log n}}$ as $n \to \infty$.

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 89 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 89: For a positive integer $n$, let $f(n)$ denote the minimum, over all sets of $n$ distinct points in the Euclidean plane $\mathbb{R}^2$, of the number of distinct pairwise distances determined by those points. Determine whether there exists a positive constant $C$ such that for all sufficiently large $n$, every set of $n$ distinct points in $\mathbb{R}^2$ determines at least $C \cdot \frac{n}{\sqrt{\log n}}$ distinct distances. Equivalently, determine whether $f(n)$ satisfies $f(n) \gg \frac{n}{\sqrt{\log n}}$ as $n \to \infty$. [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 89 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 89: For a positive integer $n$, let $f(n)$ denote the minimum, over all sets of $n$ distinct points in the Euclidean plane $\mathbb{R}^2$, of the number of distinct pairwise distances determined by those points. Determine whether there exists a positive constant $C$ such that for all sufficiently large $n$, every set of $n$ distinct points in $\mathbb{R}^2$ determines at least $C \cdot \frac{n}{\sqrt{\log n}}$ distinct distances. Equivalently, determine whether $f(n)$ satisfies $f(n) \gg \frac{n}{\sqrt{\log n}}$ as $n \to \infty$.

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 89 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 89: For a positive integer $n$, let $f(n)$ denote the minimum, over all sets of $n$ distinct points in the Euclidean plane $\mathbb{R}^2$, of the number of distinct pairwise distances determined by those points. Determine whether there exists a positive constant $C$ such that for all sufficiently large $n$, every set of $n$ distinct points in $\mathbb{R}^2$ determines at least $C \cdot \frac{n}{\sqrt{\log n}}$ distinct distances. Equivalently, determine whether $f(n)$ satisfies $f(n) \gg \frac{n}{\sqrt{\log n}}$ as $n \to \infty$.

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 89 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 89 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 89: For a positive integer $n$, let $f(n)$ denote the minimum, over all sets of $n$ distinct points in the Euclidean plane $\mathbb{R}^2$, of the number of distinct pairwise distances determined by those points. Determine whether there exists a positive constant $C$ such that for all sufficiently large $n$, every set of $n$ distinct points in $\mathbb{R}^2$ determines at least $C \cdot \frac{n}{\sqrt{\log n}}$ distinct distances. Equivalently, determine whether $f(n)$ satisfies $f(n) \gg \frac{n}{\sqrt{\log n}}$ as $n \to \infty$. [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-89`, 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 89, maintained status record. Erdős Problems record 89, checked 2026-08-01. Problem 89; status field and linked bibliography https://www.erdosproblems.com/89
   - Also cited at Problem 89; 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 Erdős Distinct Distances Lower Bound Conjecture: 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 89 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 Erdős Distinct Distances Lower Bound Conjecture: 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 89. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/89.lean:L47; theorem erdos_89; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/89.lean#L47
   - 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 Erdős Distinct Distances Lower Bound Conjecture: Supplies the pinned formal declaration whose human-readable editorial statement is published here.