Breakthrough Tracker record
A preprint claims a negative solution to the Erdős–Sárközy product-density problem
Campbell claims that no universal c>0 forces at least cM distinct same-color products below every sufficiently large M for each red-blue coloring of the positive integers; equal factors are allowed and the cutoff may depend on the coloring.
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- Stable ID
math-erdos-sarkozy-product-density-2026- Revision
math-erdos-sarkozy-product-density-2026.v1- Field
- Mathematics · Multiplicative Ramsey theory and number theory
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Provisional claimed proof
- Last checked
AI role
The author describes extensive GPT-6 Astra Pro interactions beginning October 18, followed by substantial corrections, revisions and verification by the author, who wrote the solution and accepts responsibility.
Record details
- Problem or result
- Campbell claims that no universal c>0 forces at least cM distinct same-color products below every sufficiently large M for each red-blue coloring of the positive integers; equal factors are allowed and the cutoff may depend on the coloring.
- Authors
- John M. Campbell
- Institutions
- Dalhousie University
- Result date
- Preprint submitted 1 November 2026
Why it matters
Would answer a longstanding density-uniformity question posed by Erdős and Sárközy.
Limits
The negation is about a constant uniform across colorings. Do not strengthen it to zero lower density for one universal coloring or absence of monochromatic multiplicative patterns. Submission predates the prior run, although the paper is newly discovered in this cycle; do not call it a post-November-1 submission. Unrefereed; no completed independent assessment located.
Sources
- Primary: John M. Campbell, primary preprint
Correction and revision history
- 2026-11-04 — Added provisionally with theorem scope, disclosed AI role and review limits.
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