On 3-graphs with vanishing codegree Turán density

For a $k$-uniform hypergraph (or simply $k$-graph) $F$, the codegree Turán density ${\pi }_{\mathrm{co}}\left(F\right)$ 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 $V\left(H\right)$ is contained in at least $\alpha n$ edges. In this paper, we study the problem of what 3-graphs $F$ satisfy ${\pi }_{\mathrm{co}}\left(F\right)=0$. We find that this is closely related to the uniform Turán density $$, which is the supremum over all $d$ such that there are infinitely many $F$-free $k$-graphs $H$ satisfying that any induced linear-size subhypergraph of $H$ has edge density at least $d$. We prove that, for every 3-graph $F$, ${\pi }_{\mathrm{co}}\left(F\right)=0$ implies $$. We also introduce a layered structure for 3-graphs which allows us to obtain the reverse implication: Every layered 3-graph $F$ with $$ satisfies ${\pi }_{\mathrm{co}}\left(F\right)=0$. Along the way, we answer in the negative a question of Falgas-Ravry, Pikhurko, Vaughan, and Volec [J. Lond. Math. Soc. (2) 107 (2023), 1660–1691] about whether $$ always holds. In particular, we construct counterexamples $F$ with positive but arbitrarily small ${\pi }_{\mathrm{co}}\left(F\right)$ while having $$. Our proof relies on a random geometric construction, graph distributions, Ramsey's theorem and a new formulation of the characterization of 3-graphs with vanishing uniform Turán density due to Reiher, Rödl, and Schacht [J. Lond. Math. Soc. 97 (2018), no. 1, 77–97].