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A power saving over the Spencer-Szemerédi-Trotter n^(4/3) bound for planar unit distances (Erdős Problem 90)

Theorem 1.1: there are absolute constants $0<C<\infty$ and $1\le\beta<4/3$ such that every set of $n$ points in the Euclidean plane has at most $Cn^\beta$ unordered pairs at distance $1$ , for every $n$ . This is the first improvement of the 1984 exponent $4/3$ . It leaves a gap to the lower bound $n^{1.0141}$ (Sawin) from the 2026 disproof of Erdős's $n^{1+o(1)}$ conjecture, and $\delta=4/3-\beta$ is not made explicit. Not addressed: distinct distances, higher dimensions, or other norms (for Valtr's metric $4/3$ is sharp).

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: A power saving over the Spencer-Szemerédi-Trotter n^(4/3) bound for planar unit distances (Erdős Problem 90) · Problem 90 · Unit distances: beating n^(4/3)

Confidence: Not scored

Registry verification: lean checked · announcement · partial

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: A power saving over the Spencer-Szemerédi-Trotter n^(4/3) bound for planar unit distances (Erdős Problem 90)

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 was read against the unit-distance problem as posed: it gives absolute $C$ and $1\le\beta<4/3$ with $u(n)\le Cn^\beta$ for all $n$ , a power saving with no explicit $\beta$ . Lean: lean/ComparatorChallenges/PlanarUnitDistances.json exists and its solution module OAI.Geometry.UnitDistances.Main is present at the pinned commit; the challenge is not in the formalization catalogue (formalization.yaml). The statement file was read here: theorem main asserts that there are $C,\beta$ with $0<C$ , $1\le\beta<4/3$ and $u(n)\le Cn^\beta$ for every $n$ , where $u(n)$ is the largest number of unordered pairs at Euclidean distance $1$ in an $n$ -point subset of the plane. That states the headline. Not rebuilt here. The proof combines random cuttings, entropy and prediction arguments with heights of algebraic numbers; the saving is non-effective.

Source ↗

Companion manuscript: A power saving for planar unit distances

paper · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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The weak pinned Erdős distance conjecture: some point determines n^(1-o(1)) distinct distances (Erdős Problem 604, first part)

witness · pending

paper link supplied by the VibeMathed registry.

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Lean proof: OAI/Geometry/UnitDistances/Main.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Lean statement: ComparatorChallenges/PlanarUnitDistances.lean

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Erdős Problem 90 (erdosproblems.com)

specialist registry · pending

problem-record link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean proof: OAI/Geometry/UnitDistances/Main.lean

lean artifact · pending

Artifact ↗

Lean statement: ComparatorChallenges/PlanarUnitDistances.lean

lean artifact · pending

Artifact ↗

Compute record

No linked compute attempts recorded.

Lineage and corrections

The weak pinned Erdős distance conjecture: some point determines n^(1-o(1)) distinct distances (Erdős Problem 604, first part) evidence for this event

Erdős Problem 90 (erdosproblems.com) evidence for this event

Companion manuscript: A power saving for planar unit distances evidence for this event

This event attributed to Unreleased internal OpenAI model

Lean statement: ComparatorChallenges/PlanarUnitDistances.lean evidence for this event

A power saving over the Spencer-Szemerédi-Trotter n^(4/3) bound for planar unit distances (Erdős Problem 90) parent of this event

VibeMathed record: A power saving over the Spencer-Szemerédi-Trotter n^(4/3) bound for planar unit distances (Erdős Problem 90) evidence for this event

Lean proof: OAI/Geometry/UnitDistances/Main.lean evidence for this event

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