The Tur\'an density of the tight 5-cycle minus one edge

TL;DR

This paper determines the Turán density of certain 3-uniform hypergraphs called tight cycles minus one edge for infinitely many cases, confirming a longstanding conjecture and extending previous results.

Contribution

It proves the Turán density is 1/4 for all such cycles with length not divisible by 3, confirming a conjecture and extending prior large-length results.

Findings

Turán density of $C_{ ext{ell}}^{3-}$ is 1/4 for all $ ext{ell} ot ext{ divisible by } 3$

Confirmed a conjecture of Mubayi--Sudakov--Pikhurko from 2011

Extended results to all $ ext{ell} ot ext{ divisible by } 3$

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 Tur\'an 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 Lidick\'y--Mattes--Pfender [arXiv:2409.14257].