The classical Turán problem asks: given a graph H, how many edges can an n-vertex graph have while containing no isomorphic copy of H? By viewing (k+1)-uniform hypergraphs as k-dimensional simplicial complexes, we can ask a topological version of this, first posed by Nati Linial: given a k-dimensional simplicial complex S, how many facets can an n-vertex k-dimensional simplicial complex have while containing no homeomorphic copy of S? Until recently, little was known for k greater than 2. In this talk, we give an answer for general k using dependent random choice – a method that has produced powerful results in extremal graph theory, additive combinatorics, and Ramsey theory.
Joint work with Jason Long and Bhargav Narayanan.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
