Dan
Kral
Leipzig University & MPI MiS
Turán problems in the uniform setting
Abstract.
Turán density is the minimum density that guarantees the existence of a given substructure. Almost all constructions that are known or conjectured to be extremal in the Turán density setting have highly non-uniform density distribution. This led Erdős and Sós to introduce the notion of the uniform Turán density, which requires the density of the host structure to be uniformly distributed. Erdős and Sós specifically asked to determine the uniform Turán densities of two 3-uniform hypergraphs: the Broken Tetrahedron and the Tetrahedron (K_4^{(3)-} and K_4^{(3)}). The case of the Broken Tetrahedron was resolved by Glebov, Volec and the speaker [Israel J. Math. 211 (2016), 349-366], and a computer-free proof was given by Reiher, Rödl and Schacht [J. Eur. Math. Soc. 97 (2018), 77–97]. The hypergraph regularity based approach of Reiher, Rödl and Schacht revolutionized the area and led to numerous results on the uniform Turán density of hypergraphs, which we survey during the talk. These include the spectacular result of Lamaison linking the uniform Turán densities and palette constructions, which are based on the lower bound method of Rödl. We conclude by reporting on a recent result of Kielak, Lamaison, Liu, Shu, Wu and the speaker, which resolves the Tetrahedron case and confirms that Rödl's lower bound construction from 1986 is optimal.