Tur\'{a}n density of tight cycles minus one edge in the $\ell_2$-norm

TL;DR

This paper determines the Turán density of certain tight cycles minus one edge in 3-uniform hypergraphs under the $ ext{l}_2$-norm, confirming a conjecture and showing near-extremal structures are close to a specific blowup configuration.

Contribution

It proves the Turán density for $C_{ ext{ell}}^{3-}$ in 3-graphs under the $ ext{l}_2$-norm and confirms a conjecture about the structure of near-extremal hypergraphs.

Findings

Maximum $ ext{l}_2$-norm 3-graphs avoiding $C_{ ext{ell}}^{3-}$ are close to blowups of a single triple.

Confirmed a conjecture of Balogh--Clemen--Lidický in a stronger form.

Established structural stability near the extremal configuration.

Abstract

The $3$-uniform tight $\ell$-cycle minus one edge $C_{\ell}^{3-}$ is the $3$-graph on $\ell$ vertices consisting of $\ell-1$ consecutive triples in the cyclic order. We show that for every integer $\ell \ge 5$ satisfying $\ell\not\equiv 0\pmod3$, every $C_{\ell}^{3-}$-free $3$-graph whose $\ell_2$-norm, that is, the sum of codegree squares, is close to the maximum must be structurally close to the iterative blowup of a single triple. This confirms a conjecture of Balogh--Clemen--Lidick\'{y}~[Surveys in combinatorics 2022, 21-63] in a stronger form.