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Breakthrough Tracker record

The Erdős–Ulam artifact sharpens bounds and narrows its claimed scope

Bhattacharjee, Mandal and Bhattacharya claim answers to the two questions recorded as Erdős Problem 1183. The maintained artifact now reports 11n/20 as a stronger finite lower bound and preserves a subexponential upper construction, while explicitly leaving the exact orders of the associated functions open.

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Stable ID
math-erdos-ulam-union-closed-families-2026
Revision
math-erdos-ulam-union-closed-families-2026.v2
Field
Mathematics · Extremal combinatorics, Ramsey theory and formal verification
Evidence
Tier 1 · Peer reviewed: No
Record state
Provisional · Formalization-backed provisional resolution
Last checked

AI role

The paper and artifact say Claude assisted with formal coding; they do not attribute the mathematical argument to AI.

Record details

Problem or result
Erdős–Ulam monochromatic union-closed-family problem, Erdős Problem 1183
Authors
Deep Bhattacharjee, Priyabrata Mandal and Ushashi Bhattacharya
Institutions
Formerly EGSPL, Bhubaneswar; Maulana Azad National Institute of Technology Bhopal; formerly National Taiwan University
Result date
Preprint and first artifact published October 2; corrected and sharpened artifact releases October 6–10, 2026

Why it matters

The maintained artifact materially improves a quantitative bound, clarifies which two Erdős questions are answered and exposes a layered verification boundary instead of overstating complete formalization.

Limits

This remains unrefereed. Chojecki had independently proved the subexponential side in March 2026, so that part is not novel to this artifact. Four conventional checkers cover the finite list and Lean checks the colourings-by-size component, but the mixed-level reduction is not formalized; the exact asymptotics remain open. A footnote error was corrected without changing the stated conclusion.

Sources

  1. Primary: Bhattacharjee, Mandal and Bhattacharya, Erdős–Ulam union-closed families
  2. Proof artifact: Paper, source and Lean artifact at Zenodo
  3. Formal proof: Erdős Problem 1183 Lean repository
  4. Corrected release: Erdős 1183 artifact release v2.1.1
  5. Sharpened artifact: Pinned 11n/20 verification-boundary update

Correction and revision history

  1. 2026-10-05 — Added after primary-source, scope, status, AI-role and limitation review.
  2. 2026-10-19 — Recorded the v2.1 correction, the sharper 11n/20 bound, Chojecki's earlier subexponential proof and the remaining unformalized mixed-level reduction; retained provisional status.

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