Exact deficit identity for recursive tight-cycle constructions

Definitions and mathematical statement

The tight $3$-uniform five-cycle $C_5^{(3)}$ has vertices $1,\ldots,5$ and the five edges \[ 123,\quad234,\quad345,\quad451,\quad512. \] Each edge consists of three consecutive vertices in the cyclic order; the notation suppresses braces. A simple triple system $H$ is $C_5^{(3)}$-free if no injective map from these five vertices takes all five displayed edges to edges of $H$. Additional triples among those vertices are permitted in a copy, so the condition is not induced freeness. Define $\operatorname{ex}(n,C_5^{(3)})$ as the largest edge count of such a free hypergraph on $n$ vertices and write \[ \pi(C_5^{(3)})=\lim_{n\to\infty} \frac{\operatorname{ex}(n,C_5^{(3)})}{\binom n3}. \] The limit exists, and the conjectured value is $2\sqrt3-3$.

Thus the upper-bound target is: for every $\varepsilon>0$, all sufficiently large triple systems with density above $2\sqrt3-3+\varepsilon$ contain the five-cycle. The proposed lower density comes from recursively dividing a vertex set into parts $A,B$, taking all triples with exactly two vertices in $A$ and one in $B$, and applying the same construction inside $B$. The problem asks whether the optimal limiting density is already achieved by this type of construction. The forbidden object has all five cyclic edges; deleting an edge would give a different extremal problem and a different target.

Short English statement

Determine whether the tight five-cycle has triple-system Turán density exactly two square roots of three minus three.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 20001028 (AIM-COMBINATORICS-0153), 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 33.1. Original workshop question underlying AIM-COMBINATORICS-0153. http://aimpl.org/hypergraphturan/3/.