Characterizing $3$-graphs with Degenerate Codegree Turán Density

Determining the Turán densities $π(F)$, uniform Turán densities $π_{\therefore}(F)$, and codegree Turán densities $γ(F)$ of $3$-graphs $F$ is a notoriously difficult problem in combinatorics. A landmark result by Erdős characterized the $3$-graphs whose Turán density is zero. Half a century later, Reiher, Rödl, and Schacht characterized the $3$-graphs whose uniform Turán density is zero. Recently, Ding, Lamaison, Liu, Wang, and Yang suggested a possible characterization of $3$-graphs whose codegree Turán density is zero, proved one direction of it, and conjectured the other. In this work we confirm this conjecture, completing the characterization of $3$-graphs with codegree Turán density zero. A particular difficulty of this result is that it is known that there are $3$-graphs with arbitrarily small codegree Turán density, contrasting the behavior of the normal and the uniform Turán density. Similar results were independently announced by Chen, Fang, and Yang.