Turán problem for the tetrahedron (K_4^{(3)})

Determine whether the asymptotic extremal edge density for 3-uniform hypergraphs avoiding K_4^{(3)} equals 5/9; i.e., prove that the lower bound 5/9 is sharp so that C_turan = 5/9.

Background

The Turán density for K_4{(3)} is a long-standing problem in extremal hypergraph theory. Flag algebra techniques give the best current upper bounds slightly above 0.56, while constructions yield the lower bound 5/9.

Confirming sharpness would settle a central case of hypergraph Turán problems.

References

It is conjectured that the lower bound is sharp, thus C_{\ref{turan} = \frac{5}{9}.

— Mathematical exploration and discovery at scale  (2511.02864 - Georgiev et al., 3 Nov 2025) in Subsection “Turán hypergraph problem for the tetrahedron” (Section 4.23)