The Hypergraph Turán Densities of Tight Cycles Minus an Edge

A tight $\mathrm{\ell }$-cycle minus an edge $C_{\mathrm{\ell }}^{-}$ is the $3$-graph on the vertex set $[\mathrm{\ell }]$, where any three consecutive vertices in the string $123\dots \mathrm{\ell }1$ form an edge. We show that for every $\mathrm{\ell }\ge 5$, $\mathrm{\ell }$ not divisible by $3$, the extremal number is $ex(C_{\mathrm{\ell }}^{-},n)=\frac{1}{24}{n}^3+O(n\ln n)=(\frac{1}{4}+o(1))\left(\genfrac{}{}{0px}{}{n}{3}\right)\mathrm{.}$ We determine the extremal graph up to $O(n)$ edge edits.