The Turán density of the tight 5-cycle minus one edge

Let the tight $\mathrm{\ell }$-cycle minus one edge $C_{\mathrm{\ell }}^{3-}$ be the $3$-graph on $\{1,\dots ,\mathrm{\ell }\}$ consisting of $\mathrm{\ell }-1$ consecutive triples in the cyclic order. We show that, for every $\mathrm{\ell }\ge 5$ not divisible by $3$, the Turán density of $C_{\mathrm{\ell }}^{3-}$ is $1\mathrm{/}4$ and also prove some finer structure results. This proves a conjecture of Mubayi–Sudakov–Pikhurko from 2011 and extends the results of Balogh–Luo [Combinatorica 44 (2024) 949–976] who established analogous claims for all sufficiently large $\mathrm{\ell }$. Results similar to ours were independently obtained by Lidický–Mattes–Pfender [arXiv:2409.14257].