PIMS-UVic Discrete Math Seminar: Bjarne Schülke

Since suggested by Tur\'an in 1941, determining the Tur\'an density of hypergraphs has been a notoriously difficult problem at the center of extremal combinatorics. Subsequently, several natural variants of this problem have been suggested, most prominently the uniform Tur\'an density by Erd\H{o}s and S\'os and the codegree Tur\'an density by Mubayi and Zhao. Roughly speaking, the Tur\'an density is the threshold of the edge density above which large hypergraphs are guaranteed to contain a copy of a fixed hypergraph~$F$. Similarly, the uniform Tur\'an density and the codegree Tur\'an density are the thresholds of the local density and the minimum codegree, respectively, above which large hypergraphs are guaranteed to contain a copy of~$F$. In this talk, we will discuss recent results which determine several variants of the Tur\'an density in new instances and make progress towards a problem of Erd\H{o}s and S\'os. Further, we will present our recent result with Piga which states that there are hypergraphs with arbitrarily small codegree Tur\'an density. This is in contrast to the behaviour of the classical Tur\'an density and the uniform Tur\'an density due to results by Erd\H{o}s and by Reiher, R\"odl, and Schacht, respectively.

Based on joint works with Chen, Conlon, Piga, and Sales.