[#P3024] Linear upper bound for unit distances in convex polygons

A finite mathematical diagram showing vertices of a convex polygon with unit-distance chords. — A convex polygon with selected unit-distance chords.

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 $n \to \infty$?

1Context

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.

2Problem setup

Definition 1 (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 2 (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 3 (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 1. 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.

3What 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 $n \to \infty$?

1Status

Current status (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 $n \to \infty$?^[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 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] Relevance to this problem For erdos problem 96, pins the dated research frontier: 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. 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 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 $n \to \infty$? Open full record →
	Trace	Reported
attempt · Route 1 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] Relevance to this problem For Linear upper bound for unit distances in convex polygons, record erdos-problem-96-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 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. 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 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 $n \to \infty$? 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 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.

How the 2 records connectTyped relations and evidence flow

ProblemLinear upper bound for unit distances in convex polygons

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, “Linear upper bound for unit distances in convex polygons,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-96

This page as plain text: erdos-problem-96.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 96, maintained status record. Erdős Problems record 96, checked 2026-08-01. Problem 96; 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 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.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.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 96 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 Linear upper bound for unit distances in convex polygons: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 96. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/96.lean:L69; theorem erdos_96; 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 Linear upper bound for unit distances in convex polygons: 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.

[#P3024] Linear upper bound for unit distances in convex polygons

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