Determine the Turán densities of complete and nearly complete 3-uniform hypergraphs

Determine the Turán densities of the complete 3-uniform hypergraph on four vertices, $\pi(K_4^{(3)})$, and of the 3-uniform hypergraph $K_4^{(3)-}$ obtained by deleting one edge from $K_4^{(3)}$.

Background

The paper identifies two specific unresolved instances of the classical hypergraph Turán-density problem. Turán's problem asks for the exact value of π(K4(3))\pi(K_4^{(3)}), while the related forbidden hypergraph K4(3)−K_4^{(3)-}, consisting of the complete 3-graph on four vertices with one edge removed, is described as a seemingly simpler case that nevertheless remains unresolved.

References

In particular, the problem of determining the Tur\n an density of the complete~$3$-graph on four vertices, i.e.,~$\pi(K_4{(3)})$, was asked by Tur\n an in 1941 and Erd\H{o}s offered {$500} for determining any~$\pi(K_{\ell}{(k)})$ with~$\ell > k\geq 3$ and~{$1000} for determining all~$\pi(K_{\ell}{(k)})$ with~$\ell > k\geq 3$. 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

the Turán densities $\pi(K_4{(3)-})$ and $\pi(K_4{(3)})$ remain unknown, where $K_4{(3)}$ is the complete $3$-graph on four vertices and $K_4{(3)-}$ is obtained from it by deleting one edge.

— 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