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

Abstract

Let the tight $\ell$-cycle minus one edge $C_{\ell }^{3-}$ be the $3$-graph on $\{1,\dots ,\ell \}$ consisting of $\ell -1$ consecutive triples in the cyclic order. We show that, for every $\ell \ge 5$ not divisible by $3$, the Turán density of $C_{\ell }^{3-}$ is $1/4$ 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 $\ell$. Results similar to ours were independently obtained by Lidický--Mattes--Pfender [arXiv:2409.14257].

26 pages, added anc files