A problem of Tur\'{a}n. Tur\'{a}n observed that dividing the vertices into three equal parts $X_1,X_2,X_3$, and taking the edges to be those triples that either have exactly one vertex in each part or two vertices in $X_i$ and one vertex in $X_{i+1}$ (where $X_4=X_1$) shows that\[\mathrm{ex}_3(n,K_4^3)\geq\left(\frac{5}{9}+o(1)\right)\binom{n}{3}.\]This is probably the truth. The current best upper bound is\[\mathrm{ex}_3(n,K_4^3)\leq 0.5611666\binom{n}{3},\]due to Razborov \cite{Ra10}.

See also [712] for the general case.


References


[Ra10] Razborov, Alexander A., On 3-hypergraphs with forbidden 4-vertex configurations. SIAM J. Discrete Math. (2010), 946-963.