A 3-uniform hypergraph 
on 
vertices is uniformly 
-dense if every subset 
of its vertex set satisfies
 |
where 
is the number of hyperedges contained in 
. The uniform Turán density 
of a 3-uniform hypergraph 
is the supremum
of the real numbers 
such that for every 
, there are arbitrarily large 
-free uniformly 
-dense 3-uniform hypergraphs.
Erdős and Sós (1982) introduced this parameter.
The tetrahedron is the complete 3-uniform hypergraph 
. Bucić (2026) and Kielak
et al. (2026) independently proved
 |
Bucić (2026) reports that ChatGPT-6 supplied a tensor projection identity used in his otherwise human-developed argument, and that the formal verifier Aristotle checked the identity. As of Sep. 22, 2026, external specialist review of the two papers had not been reported.
