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arXiv · 2609.08336

Solution of uniform Tur\'an's Tetrahedron Problem

Bartłomiej Kielak·Daniel Král·Ander Lamaison·Hong Liu·Xichao Shu·Zhuo Wu
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Abstract

Tur\'an's Tetrahedron Problem asks to determine the Tur\'an density of the complete hypergraph $K_4^{(3)}$ (tetrahedron). This problem, posed by Tur\'an in 1941, is one of the most famous problems in extremal combinatorics and its solution would attract \$500 prize from Erd\H{o}s. In the 1980s, Erd\H{o}s and S\'os asked to determine Tur\'an 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, Kr\'al' and Volec [Israel J. Math. 211 (2016), 349-366] and Reiher, R\"odl and Schacht [J. Eur. Math. Soc. 20 (2018), 1139-1159]. We solve the tetrahedron case by proving that the uniform Tur\'an density of $K_4^{(3)}$ is equal to 1/2; this confirms that R\"odl's lower bound construction from 1986 is optimal.

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Bartłomiej Kielak, Daniel Král, Ander Lamaison, Hong Liu, Xichao Shu, Zhuo Wu. 2026-09-08. Solution of uniform Tur\'an's Tetrahedron Problem. https://arxiv.org/abs/2609.08336

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