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 $K_4^{(3)}$.
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.