Characterizing $3$-graphs with Degenerate Codegree Turán Density

@misc{king2026characterizing,
  title = {{Characterizing $3$-graphs with Degenerate Codegree Turán Density}},
  author = {Dylan King and Simón Piga and Marcelo Sales and Bjarne Schülke},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36273},
  eprint = {2610.36273},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36273},
}

TY  - UNPB
TI  - Characterizing $3$-graphs with Degenerate Codegree Turán Density
AU  - Dylan King
AU  - Simón Piga
AU  - Marcelo Sales
AU  - Bjarne Schülke
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.36273
UR  - https://arxiv.org/abs/2610.36273
AB  - 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.
ER  -