Turán density of the complete 3-graph on four vertices

Determine the Turán density \(\pi(K_4^{(3)})\) of the complete 3-uniform hypergraph on four vertices.

Background

The complete 3-uniform hypergraph K4(3)K_4^{(3)} is identified as a canonical unresolved case of the general hypergraph Turán-density problem. The paper states that the problem was posed by Turán in 1941 and notes Erdős's monetary prize for determining such densities, emphasizing its longstanding status.

References

Despite receiving a lot of attention (see, for instance, the surveys on the topic~\cites{F:91,K:11,S:95}), this problem, and even the seemingly simpler problem of determining~$\pi(K_4{(3)-})$, where~$K_4{(3)-}$ is the~$K_4{(3)}$ minus one edge, remain open.

— Infinitely many accumulation points of codegree Turán densities  (2502.13485 - Li et al., 19 Feb 2025) in Introduction