Seminar Details

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 > 2. In this talk, we give an answer for general k, by way of dependent random choice and the combinatorial notion of a trace-bounded hypergraph. Joint work with Jason Long and Bhargav Narayanan.