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3. Tight $5$-Cycle

Note to the editors: The items in this section are in the wrong order. The next version of the code will provide an easy way to reorder the problems in a section.
    1.     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.

      Conjecture 3.1.

        • Tight $5$-Cycle Minus an Edge

              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.

          Conjecture 3.2.

          [Mubayi and Pikhurko] $\pi(C_5^-)=1/4$.

              Cite this as: AimPL: Hypergraph Turán problem, available at http://aimpl.org/hypergraphturan.