[#P3020] Erdős Distinct Distances Lower Bound Conjecture

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 $n \to \infty$.

A finite mathematical diagram showing planar points joined by several distinct distances. — Planar points joined by several distinct distances.

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Definitions and notation

1Context

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.

2Definitions

Definition 1 (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 2 (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 3 (The natural logarithm). The natural logarithm is denoted by $\log n$, and $\sqrt{\log n} = (\log n)^{1/2}$.

3What 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 $n \to \infty$.

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 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]

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 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 $n \to \infty$.	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 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] Relevance to this problem For erdos problem 89, pins the dated research frontier: 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. 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 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 $n \to \infty$. Open full record →
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 $n \to \infty$.	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 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] Relevance to this problem For Erdős Distinct Distances Lower Bound Conjecture, record erdos-problem-89-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 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. 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 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 $n \to \infty$. 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 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.

How the 2 records connect

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

ProblemErdős Distinct Distances Lower Bound Conjecture

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

“Erdős Distinct Distances Lower Bound Conjecture.” TheoremDB. P3020. Problem statement; statement text SHA-256 f91bad5dfb9cefa65acbdfb9dee7464f0c0cf01c581d6fca4a12bd7435a8d6ea. https://theoremdb.org/statement/?ref=P3020

BibTeX

@misc{theoremdb-problem-f91bad5dfb9cefa65acbdfb9dee7464f0c0cf01c581d6fca4a12bd7435a8d6ea,
  title = {{Erdős Distinct Distances Lower Bound Conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 f91bad5dfb9cefa65acbdfb9dee7464f0c0cf01c581d6fca4a12bd7435a8d6ea},
  url = {https://theoremdb.org/statement/?ref=P3020}
}

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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 89, maintained status record. Erdős Problems record 89, checked 2026-08-01. Problem 89; 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 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.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.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 89 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 Distinct Distances Lower Bound Conjecture: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 89. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/89.lean:L47; theorem erdos_89; 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 Distinct Distances Lower Bound Conjecture: 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.

[#P3020] Erdős Distinct Distances Lower Bound Conjecture

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