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

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 $n \to \infty$?

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

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Definitions and notation

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.

Remark 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 2 (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 $n \geq N$.

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

Saved packet · July 31, 2026

What counts as a solution

Saved packet 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

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Record	Kind	Assessment
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 $n \to \infty$?	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
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 →
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 $n \to \infty$?	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
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 material

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 connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemLinear upper bound for unit distances in convex polygons

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Grid points with no three collinear and no four concyclicdiscrete geometry
Largest equilateral-triangle-free subset of the fifteen-row triangular latticediscrete geometry
Most lattice points with all pairwise slopes distinct in a ten by ten griddiscrete geometry

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Plain text

“Linear upper bound for unit distances in convex polygons.” TheoremDB. P3024. Problem statement; statement text SHA-256 9001db3de2353249ae834700427c9d9d229cefa6b8ea7bee7c05121746330223. https://theoremdb.org/statement/?ref=P3024

BibTeX

@misc{theoremdb-problem-9001db3de2353249ae834700427c9d9d229cefa6b8ea7bee7c05121746330223,
  title = {{Linear upper bound for unit distances in convex polygons}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 9001db3de2353249ae834700427c9d9d229cefa6b8ea7bee7c05121746330223},
  url = {https://theoremdb.org/statement/?ref=P3024}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

1Lean verification

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

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