Turán's problem for hypergraphs

Definitions and mathematical statement

A finite simple 3-uniform hypergraph is a pair $H=(V,E)$ with $E\subseteq\binom V3$: each edge is an unordered triple of distinct vertices and is counted once. It contains a complete 3-uniform hypergraph on $s$ vertices if some $S\subseteq V$, $|S|=s$, has all $\binom s3$ triples as edges. Write $K_s^{(3)}$ for this complete hypergraph.

The record asks for the following two finite extremal assertions, for every integer $n\ge1$: \[ \begin{aligned} |V|=3n, K_4^{(3)}\not\subseteq H &\quad\Longrightarrow\quad |E|\le\tfrac12n^2(5n-3),\\ |V|=2n, K_5^{(3)}\not\subseteq H &\quad\Longrightarrow\quad |E|\le n^2(n-1). \end{aligned} \] The forbidden copies are ordinary subhypergraphs, not induced copies; for complete hypergraphs these descriptions coincide on the selected vertex set.

The proposed values have concrete witnesses. In the first problem partition $V=X\sqcup Y\sqcup Z$ with each part of size $n$, and include triples of types $XYZ$, $XXY$, $YYZ$, and $ZZX$. Their number is $n^3+3\binom n2n=\tfrac12n^2(5n-3)$, and no four vertices have all four triples present. In the second, partition into two parts of size $n$ and include precisely the triples meeting both parts. Their number is $2n\binom n2=n^2(n-1)$; every five vertices contain three in one part, whose triple is absent. The target is optimality of these exact counts at the displayed orders, not only their limiting densities.

Short English statement

Are the stated balanced constructions extremal for triple systems containing no complete four-vertex, respectively five-vertex, triple system?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3296 (OPG-47343), OpenGarden collection. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Turán's problem for hypergraphs. Original problem and discussion; the mathematical formulation below is expanded from the supplied source record. http://www.openproblemgarden.org/op/turans_problem_for_hypergraphs.