Solution of uniform Turán's Tetrahedron Problem

@misc{kielak2026solution,
  title = {{Solution of uniform Turán's Tetrahedron Problem}},
  author = {Bartłomiej Kielak and Daniel Král and Ander Lamaison and Hong Liu and Xichao Shu and Zhuo Wu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.08336},
  eprint = {2609.08336},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.08336},
}

TY  - UNPB
TI  - Solution of uniform Turán's Tetrahedron Problem
AU  - Bartłomiej Kielak
AU  - Daniel Král
AU  - Ander Lamaison
AU  - Hong Liu
AU  - Xichao Shu
AU  - Zhuo Wu
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.08336
UR  - https://arxiv.org/abs/2609.08336
AB  - Turán's Tetrahedron Problem asks to determine the Turán density of the complete hypergraph K_4^(3) (tetrahedron). This problem, posed by Turán in 1941, is one of the most famous problems in extremal combinatorics and its solution would attract $500 prize from Erdős. In the 1980s, Erdős and Sós asked to determine Turán densities of K_4^(3)− (broken tetrahedron) and K_4^(3) (tetrahedron) when edges are constrained to be uniformly distributed in the host hypergraph. The presumably easier case of the broken tetrahedron was solved by Glebov, Král' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, Rödl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159]. We solve the tetrahedron case by proving that the uniform Turán density of K_4^(3) is equal to 1/2; this confirms that Rödl's lower bound construction from 1986 is optimal.
ER  -