Uniform Turán density of the three-uniform tetrahedron
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What is the uniform Turán density of the three-uniform tetrahedron K₄³? The uniform notion requires edge density throughout large vertex subsets, rather than only a global edge count.
Why it matters and what counts as progress
Why it matters: The threshold identifies when broadly distributed density forces a tetrahedron.
What counts as progress: A matching upper bound of 1/2 for the known lower construction, or an alternative rigorous certificate proving that same bound under the uniform-density definition.
How the problem developed
- 1982 · Open problem [1]
Paul Erdős and Vera T. Sós Asked for the uniform Turán density of the three-uniform tetrahedron, in the problem line introduced for uniformly distributed hypergraphs.
- 9 Sept 2026 · Reported result [1]
Bartłomiej Kielak, Daniel Král’, Ander Lamaison, Hong Liu, Xichao Shu and Zhuo Wu Reported that the uniform Turán density of K₄³ equals 1/2, matching Rödl’s 1986 lower-bound construction.
- 10 Sept 2026 · Independent proof [2]
Matija Bucić Posted an independent certificate-based proof of the same uniform density value.
AI contribution
The uniform Turán density of the tetrahedron
In Bucić’s alternative proof that the uniform Turán density of the three-uniform tetrahedron K₄³ is 1/2, ChatGPT 6 Pro found the finite sum-of-positive-quadratic-forms identity and proposed the tensor projection inequality used in the proof.
Sources
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