Hypergraph Turán problem

    Mubayi and Rödl [mubayi+rodl:02] have given bounds on $\pi(C_5^3)$, where $C_5^3$ is the tight $3$-graph $5$-cycle: $$ C_5^3=\{123,234,345,451,512\}. $$ In particular, the lower bound $\pi(C_5^3)\ge 2\sqrt3-3$ comes from the following construction: partition the vertex set into two parts $A$ and $B$, take all triples that intersect $A$ precisely in $2$ vertices, and recursively repeat this construction within $B$. Finding the optimal ratio between $|A|$ and $|B|$ gives the required. Razborov’s [razborov:10] flag algebra computations showed that $\pi(C_5^3)< 0.4683$ (note that $2\sqrt3-3=0.4641...$). This makes the following conjecture plausible.

    Let the $3$-graph $C_5^-$ be obtained from $C_5^3$ by removing one edge. An example of a $C_5^-$-free $3$-graph can be obtained by taking a complete $3$-partite $3$-graph and repeating this construction recursively within each of the three parts. This gives density $1/4$ in the limit.