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Breakthrough Tracker record

Two independent preprints claim uniform Turán density one half for the tetrahedron

Kielak and collaborators and, independently, Matija Bucić claim that the uniform Turán density of the complete 3-uniform tetrahedron equals one half. The papers use distinct combinatorial and Fourier-analytic routes.

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Stable ID
math-uniform-turan-tetrahedron-density-2026
Revision
math-uniform-turan-tetrahedron-density-2026.v2
Field
Mathematics · Extremal combinatorics, hypergraphs and computer-assisted proof
Evidence
Tier 1 · Peer reviewed: No
Record state
Provisional · Independently corroborated preprint claim
Last checked

AI role

Kielak et al. say no presented argument came from AI and that an earlier AI suggestion was superseded. Bucić discloses a substantive ChatGPT 6 Pro contribution to the key identity and certificate.

Record details

Problem or result
Uniform Turán density of the complete 3-uniform tetrahedron
Authors
Bartłomiej Kielak, Daniel Král’, Ander Lamaison, Hong Liu, Xichao Shu and Zhuo Wu; independently, Matija Bucić
Institutions
Leipzig University; MPI MiS; Public University of Navarre; Institute for Basic Science; UPC
Result date
Independent preprints revised through September 24, 2026

Why it matters

The uniform problem is a long-open extremal-hypergraph question and fixes its exact density. It must not be confused with the original unrestricted 1941 Turán tetrahedron problem, which remains open.

Limits

Both proof routes use supplied exact or computer-checkable certificates and neither is peer reviewed. Independent convergence is strong support but not journal acceptance or a complete outside audit; the unrestricted 1941 Turán problem remains open.

Sources

  1. Primary: Kielak et al., uniform Turán tetrahedron solution
  2. Independent preprint: Bucić, Fourier-analytic proof of the uniform Turán density

Correction and revision history

  1. 2026-09-10 — Added after primary-source, scope, status, AI-role and limitation review.
  2. 2026-10-02 — Added Bucić’s independent Fourier-analytic preprint and the v3/v2 histories; provisional status retained.

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