An alternating flat family for the complete 3-graph lower bound

Definitions and mathematical statement

A simple $3$-uniform hypergraph on $N$ vertices is a collection of three-element subsets of an $N$-element vertex set. For an integer $m\geq4$, write $K_m^{(3)}$ for the complete $3$-uniform hypergraph on $m$ vertices. A hypergraph is $K_m^{(3)}$-free if no $m$ of its vertices support all $\binom m3$ triples. Define the extremal number and density \[ \operatorname{ex}(N,K_m^{(3)}) =\max\{|E(H)|:|V(H)|=N,\ H\text{ is }K_m^{(3)}\text{-free}\}, \qquad \pi(K_m^{(3)})=\lim_{N\to\infty} \frac{\operatorname{ex}(N,K_m^{(3)})}{\binom N3}. \] The limit exists by the usual averaging monotonicity for normalized extremal numbers. The problem is to prove or disprove the proposed formula, for every fixed $m\geq4$, \[ \pi(K_m^{(3)})=1-\frac{4}{(m-1)^2}. \] Concretely, the upper-bound direction says that for every $\varepsilon>0$ there is $N_0(m,\varepsilon)$ such that a hypergraph of density above $1-4/(m-1)^2+\varepsilon$ contains $K_m^{(3)}$ whenever $N\geq N_0$. Sharpness requires arbitrarily large forbidden-clique-free hypergraphs whose densities approach the proposed value from below. The clique size remains fixed while the vertex count tends to infinity. Edges are unordered triples without repeated vertices, and neither ordered hypergraphs nor induced-forbidden-subgraph conventions are involved.

Short English statement

Determine whether the clique Turán density for every fixed three-uniform clique equals the proposed $1-4/(m-1)^2$.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001022 (AIM-COMBINATORICS-0147), 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 11.2. Original workshop question underlying AIM-COMBINATORICS-0147. http://aimpl.org/hypergraphturan/1/.