Uniform Turán density of hypergraphs
Organizer
Benjamin Sudakov
Speaker
Daniel Král
Time
Tuesday, May 28, 2024 5:05 PM - 6:15 PM
Venue
Online
Online
Zoom 787 662 9899
(BIMSA)
Abstract
In the early 1980s, Erdős and Sós, initiated the study of the classical Turán problem with a uniformity condition: the uniform Turán density of a hypergraph H is the infimum over all d for which any sufficiently large hypergraph with the property that all its linear-size subhypergraphs have density at least d contains H. In particular, they raised the questions of determining the uniform Turán densities of $K_4^3$, the complete $4$-vertex $3$-uniform hypergraph, and $K_4^3$, the hypergraph $K_4^3$ with an edge removed. The latter question was solved only recently in [Israel J. Math. 211 (2016), 349–366] and [J. Eur. Math. Soc. 97 (2018), 77–97], while the former still remains open for almost 40 years. In this talk, we survey recent and some very recent results concerning the uniform Turán density of hypergraphs and particularly focus on constructions of hypergraphs with various values of uniform Turán density.
The talk is based on results obtained jointly with (subsets of) Matija Bucić, Jacob W. Cooper, Frederik Garbe, Daniel Iľkovič, Filip Kučerák, Ander Lamaison, Samuel Mohr and David Munhá Correia.
The talk is based on results obtained jointly with (subsets of) Matija Bucić, Jacob W. Cooper, Frederik Garbe, Daniel Iľkovič, Filip Kučerák, Ander Lamaison, Samuel Mohr and David Munhá Correia.
Speaker Intro
Daniel Kráľ is a Czech scientist who works on a range of topics in mathematics and computer science and at their interface, where he develops and applies mathematical methods to address problems of interest in computer science. Professor Kráľ obtained his PhD in 2004 from Charles University in Prague and subsequently held positions at TU Berlin, the Georgia Institute of Technology, Charles University and the University of Warwick. Currently, he is the inaugural holder of the Donald Ervin Knuth Professorship in Computer Science at Masaryk University in Brno and an honorary professor of Computer Science and Mathematics at the University of Warwick. Professor Kráľ received the European Prize in Combinatorics in 2011 and was appointed a Fellow of the American Mathematical Society in 2020 and a SIAM Fellow in 2024.