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

For a k-uniform hypergraph (or simply k-graph) F, the codegree Tur\'{a}n density πco(F) is the supremum over all α 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 α n edges. Recently, it was proved that for every $3$-graph F, πco(F)=0 implies π∴(F)=0, where π∴(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 π∴(F)=0 satisfies πco(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 π∴(F)≤πco(F) always holds. In particular, we construct counterexamples F with positive but arbitrarily small πco(F) while having π∴(F)≥ 4/27.