Uniform Turn density of hypergraphs | Research seminar in Discrete Mathematics

Abstract

In the early 1980s, Erdos and Ss, initiated the study of the classical Turn problem with auniformity condition: the uniform Turn density of a hypergraph H is the infimum over all d for whichany sufficiently large hypergraph with the property that all its linear-size subhypergraphs havedensity at least d contains H. In particular, they raised the questions of determining the uniformTurn densities of K3, the complete 4-vertex 3-uniform hypergraph, and K3, the hypergraph K;with an edge removed. The latter guestion was solved only recently in [lsrael j. Math. 211 (2016)349366] and [. Eur. Math. Soc. 97 (2018), 7797], while the former still remains open for almost 40years. in this talk, we survey recent and some very recent results concerning the uniform Turndensity of hypergraphs and particularly focus on constructions of hypergraphs with various valuesof uniform Turn density.

The talk is based on results obtained jointly with (subsets of) Matija Bucic, Jacob W. Cooper.Frederik Garbe, Daniel llkovic, Filip Kucerk, Ander Lamaison, Samuel Mohr and David MunhCorreia.

Speaker Intro

Daniel Krl is a Czech scientist who works on a range of topics in mathematics and computerscience and at their interface, where he develops and applies mathematical methods to addressproblems of interest in computer science. Professor Krf' obtained his PhD in 2004 from CharlesUniversity in Prague and subseguently held positions at TU Berlin, the Georgia Institute ofFechnology, Charles University and the University of Warwick. Currently, he is the inaugural holdeof the Donald Ervin Knuth Professorship in Computer Science at Masaryk University in Brno and anhonorary professor of Computer Science and Mathematics at the University of Warwick. ProfessoKr' received the European Prize in Combinatorics in 2011 and was appointed a Fellow of theAmerican Mathematical Society in 2020 and a SlAM Fellow in 2024.