The Hypergraph Turán Densities of Tight Cycles Minus an Edge

Abstract

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