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

Since our results allow a transference of results from the dense to the sparse setting, but even in the dense setting many challenging extremal combinatorics questions remain open (such as the famous Turán problem for the $4$-vertex clique in $3$-uniform hypergraphs), it is not surprising that we cannot resolve the sparse analogues of such problems.

— A generalised transference principle  (2608.17982 - Allen et al., 18 Aug 2026) in Section 1, subsection “Applications”

Even the density of the tetrahedron $K_4{(3)}$, posed by Turan in 1941, remains unknown.

— Turing universality, computability, and incompleteness in hypergraph Turán theory  (2609.04295 - Li et al., 3 Sep 2026) in Introduction