Turán density of the complete 3-graph with one edge deleted

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

Background

The hypergraph K4(3)−K_4^{(3)-} is the complete 3-uniform hypergraph on four vertices with one edge removed. The paper describes determining its Turán density as seemingly simpler than determining π(K4(3))\pi(K_4^{(3)}), but explicitly states that it nevertheless remains unresolved.

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

Frankl and F"uredi conjectured that $\pi(H_33)=2/7$, and Gunderson and Semeraro proved that $\pi(H_34)=1/4$.

— Near-optimal Turán densities of $r$-graphs on $r+1$ vertices  (2608.18924 - Yang et al., 19 Aug 2026) in Section 1, Introduction and main results