Unique balance and a codegree-variance obstruction for the tetrahedron conjecture

Definitions and mathematical statement

A simple $3$-uniform hypergraph $H$ on $n$ vertices has an edge set $E(H)\subseteq\binom{V(H)}3$. The tetrahedron $K_4^{(3)}$ consists of all four triples on four vertices. A copy in $H$ means four distinct vertices whose four triples all belong to $E(H)$. Let $\operatorname{ex}(n,K_4^{(3)})$ be the maximum number of edges in a tetrahedron-free $3$-uniform hypergraph on $n$ vertices. Its Turán density is \[ \pi(K_4^{(3)})=\lim_{n\to\infty} \frac{\operatorname{ex}(n,K_4^{(3)})}{\binom n3}. \] The limit exists by the usual averaging monotonicity of normalized extremal densities. The conjecture asks whether this limit is $5/9$.

In upper-bound form, for every $\varepsilon>0$ and all sufficiently large $n$, every $3$-uniform hypergraph with more than $(5/9+\varepsilon)\binom n3$ edges should contain a tetrahedron. The candidate density has a concrete interpretation: partition the vertices into three nearly equal parts $A,B,C$; include triples meeting all three parts, and triples of types $AAB$, $BBC$, and $CCA$. This construction is tetrahedron-free and has density tending to $5/9$. The problem is whether any construction can have a larger limiting density. The forbidden copy is an ordinary subhypergraph, with no induced-copy convention or restrictions on vertex degrees.

Short English statement

Is five ninths the maximum limiting edge density of a tetrahedron-free triple system?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001021 (AIM-COMBINATORICS-0146), 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 11.1. Original workshop question underlying AIM-COMBINATORICS-0146. http://aimpl.org/hypergraphturan/1/.