# P3024: Linear upper bound for unit distances in convex polygons

- ID: `P3024`
- Reference: `erdos-problem-96`
- Page: https://theoremdb.org/statements/P3024
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 96: For each integer $n \geq 3$, let $U_c(n)$ denote the maximum number of unit-distance pairs among all sets of $n$ points in $\mathbb{R}^2$ that form the vertices of a convex polygon. Here, a unit-distance pair means a pair of points at Euclidean distance exactly $1$. Does there exist a constant $C > 0$ such that $U_c(n) \leq C \cdot n$ for all sufficiently large $n$? Equivalently, is it true that $U_c(n) = O(n)$ as $n \to \infty$?

### Context

This problem belongs to discrete geometry and extremal combinatorics, concerning the distribution of unit distances among points in convex position. It is related to broader questions about unit distance graphs and Erdős's famous problems on distances in point sets.

### Problem setup

- **Definition (A set of points in $\mathbb{R}^2$).** A set of points in $\mathbb{R}^2$ is in convex position, or forms a convex polygon, if each point is an extreme point of the convex hull of the set; equivalently, no point lies in the convex hull of the others.
- **Definition (For a finite set $P \subset \mathbb{R}^2$, the unit-distance pairs are the unordered pairs $\{p, q\} \subseteq P$ with $p \neq q$ such that the Euclidean distance $\|p - q\| = 1$).** For a finite set $P \subset \mathbb{R}^2$, the unit-distance pairs are the unordered pairs $\{p, q\} \subseteq P$ with $p \neq q$ such that the Euclidean distance $\|p - q\| = 1$.
- **Definition (The notation $f(n) = O(g(n))$ as $n \to \infty$).** The notation $f(n) = O(g(n))$ as $n \to \infty$ means that there exist constants $C > 0$ and $N$ such that $|f(n)| \leq C \cdot g(n)$ for all $n \geq N$.
- **Remark.** This problem belongs to discrete geometry and extremal combinatorics, concerning the distribution of unit distances among points in convex position. It is related to broader questions about unit distance graphs and Erdős's famous problems on distances in point sets.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 96: For each integer $n \geq 3$, let $U_c(n)$ denote the maximum number of unit-distance pairs among all sets of $n$ points in $\mathbb{R}^2$ that form the vertices of a convex polygon. Here, a unit-distance pair means a pair of points at Euclidean distance exactly $1$. Does there exist a constant $C > 0$ such that $U_c(n) \leq C \cdot n$ for all sufficiently large $n$? Equivalently, is it true that $U_c(n) = O(n)$ as $n \to \infty$?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 96 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 96: For each integer $n \geq 3$, let $U_c(n)$ denote the maximum number of unit-distance pairs among all sets of $n$ points in $\mathbb{R}^2$ that form the vertices of a convex polygon. Here, a unit-distance pair means a pair of points at Euclidean distance exactly $1$. Does there exist a constant $C > 0$ such that $U_c(n) \leq C \cdot n$ for all sufficiently large $n$? Equivalently, is it true that $U_c(n) = O(n)$ 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 96 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 96: For each integer $n \geq 3$, let $U_c(n)$ denote the maximum number of unit-distance pairs among all sets of $n$ points in $\mathbb{R}^2$ that form the vertices of a convex polygon. Here, a unit-distance pair means a pair of points at Euclidean distance exactly $1$. Does there exist a constant $C > 0$ such that $U_c(n) \leq C \cdot n$ for all sufficiently large $n$? Equivalently, is it true that $U_c(n) = O(n)$ as $n \to \infty$?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 96 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 96: For each integer $n \geq 3$, let $U_c(n)$ denote the maximum number of unit-distance pairs among all sets of $n$ points in $\mathbb{R}^2$ that form the vertices of a convex polygon. Here, a unit-distance pair means a pair of points at Euclidean distance exactly $1$. Does there exist a constant $C > 0$ such that $U_c(n) \leq C \cdot n$ for all sufficiently large $n$? Equivalently, is it true that $U_c(n) = O(n)$ 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 96 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 96 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 96: For each integer $n \geq 3$, let $U_c(n)$ denote the maximum number of unit-distance pairs among all sets of $n$ points in $\mathbb{R}^2$ that form the vertices of a convex polygon. Here, a unit-distance pair means a pair of points at Euclidean distance exactly $1$. Does there exist a constant $C > 0$ such that $U_c(n) \leq C \cdot n$ for all sufficiently large $n$? Equivalently, is it true that $U_c(n) = O(n)$ 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-96`, 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 96, maintained status record. Erdős Problems record 96, checked 2026-08-01. Problem 96; status field and linked bibliography https://www.erdosproblems.com/96
   - Also cited at Problem 96; 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 Linear upper bound for unit distances in convex polygons: 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 96 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 Linear upper bound for unit distances in convex polygons: 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 96. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/96.lean:L69; theorem erdos_96; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/96.lean#L69
   - 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 Linear upper bound for unit distances in convex polygons: Supplies the pinned formal declaration whose human-readable editorial statement is published here.