Uniform Tur\'an density of Tetrahedron

Speaker: Zhuo Wu，Universitat Politècnica de Catalunya/ Freie Universität Berlin            

Title: Uniform Tur\'an density of Tetrahedron

Language: Chinese 

Time & Venue: 2026.09.11  15:00-16:00   南楼N613

Abstract: Tur\'an's Tetrahedron Problem, posed in 1941, asks for the Tur\'an density of the complete 3-uniform hypergraph $K_4^{(3)}$, and remains one of the central open problems in extremal combinatorics. In the 1980s, Erd\H{o}s and S\'os proposed a uniformly distributed version of the problem, asking for the maximum density of a uniformly dense 3-uniform hypergraph avoiding $K_4^{(3)}$. While the corresponding problem for the broken tetrahedron $K_4^{(3)-}$ was settled about a decade ago, the tetrahedron case remained open. We solve this problem by showing that the uniform Tur\'an density of $K_4^{(3)}$ is $1/2$, confirming that a construction of R\"odl from 1986 is optimal. This work is joint with Bart{\l}omiej Kielak, Daniel Kr\'al', Ander Lamaison, Hong Liu, and Xichao Shu.