Turán density of tight cycles minus one edge in the ${\ell }_2$-norm

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 $\ell \not\equiv 0(\mathrm{mod}3)$, 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--Lidický~[Surveys in combinatorics 2022, 21-63] in a stronger form.

18 pages, comments are welcome. arXiv admin note: text overlap with arXiv:2412.21011, arXiv:2506.03223