Exact recursion and local profile stability for the broken tetrahedron construction

Definitions and mathematical statement

Let $F=K_4^-$ be the simple $3$-uniform hypergraph on $\{1,2,3,4\}$ with edges $123$, $124$, and $134$, where, for example, $123$ means $\{1,2,3\}$. Thus $F$ is a complete $3$-uniform hypergraph on four vertices with one triple deleted. A $3$-uniform hypergraph $H$ is $F$-free if it has no injective copy of these three edges. The missing fourth triple is unrestricted in a copy, so four vertices spanning all four triples also violate $F$-freeness. Equivalently, every set of four vertices of an $F$-free hypergraph spans at most two edges.

For $n\ge4$, set \[ \operatorname{ex}(n,F)=\max\{|E(H)|:|V(H)|=n, H\text{ is }F\text{-free}\}, \qquad \pi(F)=\lim_{n\to\infty}\frac{\operatorname{ex}(n,F)}{\binom n3}. \] The maximum ranges over simple hypergraphs; repeated edges are not counted. The normalized extremal densities have a limit by averaging over vertex subsets. The workshop conjecture is $\pi(F)=2/7$. Accordingly, a resolution requires the sharp asymptotic extremal density, not an exact formula for every finite $n$. The proposed upper bound says that for every $\varepsilon>0$, all sufficiently large triple systems of density exceeding $2/7+\varepsilon$ have four vertices spanning at least three triples. The lower side asks for $F$-free systems approaching density $2/7$.

Short English statement

Is two sevenths the Turán density of a tetrahedron with one triple removed?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001025 (AIM-COMBINATORICS-0150), AIM Workshop Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 American Institute of Mathematics, Hypergraph Turan problem, item 11.4. Original workshop question underlying AIM-COMBINATORICS-0150. http://aimpl.org/hypergraphturan/1/.