[#P3030] Erdős's distinct distances diameter problem

Problem. Erdős Problem 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$?

A finite mathematical diagram showing a planar point set with separated distance classes. — A planar point set with several distance classes marked.

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1Context

This problem originates from Paul Erdős's collection of open problems in discrete geometry. It concerns the relationship between the combinatorial condition of separated distances and the geometric size of a point set. Related results include: Kanold proved a lower bound of order $n^{3/4}$; Guth and Katz improved this to order $n / \log n$ using their solution to the Erdős distinct distances problem; and Piepmeyer constructed a 9-point example with diameter less than 5. A stronger conjecture asserts that $\operatorname{diam}(A) \geq n - 1$ for all sufficiently large $n$.

2Definitions

Definition 1 (A finite set $A \subset \mathbb{R}^2$). A finite set $A \subset \mathbb{R}^2$ is distance-separated if for all points $p_1, q_1, p_2, q_2 \in A$, whenever $\operatorname{dist}(p_1, q_1) \neq \operatorname{dist}(p_2, q_2)$, we have $|\operatorname{dist}(p_1, q_1) - \operatorname{dist}(p_2, q_2)| \geq 1$, where $\operatorname{dist}$ denotes the Euclidean distance.

Definition 2 (The diameter of a set $A \subset \mathbb{R}^2$, denoted $\operatorname{diam}(A)$). The diameter of a set $A \subset \mathbb{R}^2$, denoted $\operatorname{diam}(A)$, is the supremum of Euclidean distances between pairs of points in $A$, i.e., $\operatorname{diam}(A) = \sup\{\operatorname{dist}(p,q) : p, q \in A\}$.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$?

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 100 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$?^[1]

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 100 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$?	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 100 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$?^[1] Relevance to this problem For erdos problem 100, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 100 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 100. 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 100 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 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$? Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$?	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 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$?^[1] Relevance to this problem For Erdős's distinct distances diameter problem, record erdos-problem-100-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 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. 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 100: Let a finite set of points in the plane be called distance-separated if whenever two distances determined by pairs of its points are unequal, they differ by at least 1. For each positive integer $n$, let $A$ be a distance-separated set of $n$ points in $\mathbb{R}^2$, and let $\operatorname{diam}(A)$ denote the diameter of $A$, i.e., the maximum Euclidean distance between any two points of $A$. Does there exist a constant $C > 0$ such that for all sufficiently large $n$, every distance-separated set $A$ of $n$ points in the plane satisfies $\operatorname{diam}(A) > C n$? 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 100 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 100 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.

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

ProblemErdős's distinct distances diameter problem

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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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“Erdős's distinct distances diameter problem.” TheoremDB. P3030. Problem statement; statement text SHA-256 f58aeb5d19d4cffa53237d994381edcb8e4fdf550f35f9527055f90e08efcd86. https://theoremdb.org/statement/?ref=P3030

BibTeX

@misc{theoremdb-problem-f58aeb5d19d4cffa53237d994381edcb8e4fdf550f35f9527055f90e08efcd86,
  title = {{Erdős's distinct distances diameter problem}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 f58aeb5d19d4cffa53237d994381edcb8e4fdf550f35f9527055f90e08efcd86},
  url = {https://theoremdb.org/statement/?ref=P3030}
}

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

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1References

Packet source. Erdős Problems database, Problem 100, maintained status record. Erdős Problems record 100, checked 2026-08-01. Problem 100; 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 100; 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 Erdős's distinct distances diameter problem: 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 100 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 Erdős's distinct distances diameter problem: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 100. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/100.lean:L41; theorem erdos_100; 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 Erdős's distinct distances diameter problem: 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.

[#P3030] Erdős's distinct distances diameter problem

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