On \(3\)-graphs with vanishing codegree Tur\'{a}n density

• Published in: arXiv.org 2024-07
• Main Authors: Ding, Laihao, Ander Lamaison, Liu, Hong, Wang, Shuaichao, Yang, Haotian
• Format: Article
• Language: English
• Subjects: Density; Graph theory; Graphs
• ISSN: 2331-8422

For a \(k\)-uniform hypergraph (or simply \(k\)-graph) \(F\), the codegree Tur\'{a}n density \(\pi_{\mathrm{co}}(F)\) is the supremum over all \(\alpha\) such that there exist arbitrarily large \(n\)-vertex \(F\)-free \(k\)-graphs \(H\) in which every \((k-1)\)-subset of \(V(H)\) is contained in at least \(\alpha n\) edges. Recently, it was proved that for every \(3\)-graph \(F\), \(\pi_{\mathrm{co}}(F)=0\) implies \(\pi_{\therefore}(F)=0\), where \(\pi_{\therefore}(F)\) is the uniform Tur\'{a}n density of \(F\) and is defined as the supremum over all \(d\) such that there are infinitely many \(F\)-free \(k\)-graphs \(H\) satisfying that any induced linear-size subhypergraph of \(H\) has edge density at least \(d\). In this paper, we introduce a layered structure for \(3\)-graphs which allows us to obtain the reverse implication: every layered \(3\)-graph \(F\) with \(\pi_{\therefore}(F)=0\) satisfies \(\pi_{\mathrm{co}}(F)=0\). Along the way, we answer in the negative a question of Falgas-Ravry, Pikhurko, Vaughan and Volec [J. London Math. Soc., 2023] about whether \(\pi_{\therefore}(F)\leq\pi_{\mathrm{co}}(F)\) always holds. In particular, we construct counterexamples \(F\) with positive but arbitrarily small \(\pi_{\mathrm{co}}(F)\) while having \(\pi_{\therefore}(F)\ge 4/27\).