MIT Combinatorics Seminar

The Hypergraph Turan Problem

Peter Keevash (California Institute of Technology)

Wednesday, April 12, 2006   4:30 PM   Building 2 Room 105 


A central problem of extremal combinatorics is to determine the Tur\'an number of a given $r$-uniform hypergraph $\mathcal{F}$, i.e. the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices that does not contain a copy of $\mathcal{F}$. Since the problem was introduced over sixty years ago, it has only been solved for relatively few hypergraphs $\mathcal{F}$. Many of these results were found very recently by means of the stability method, which has brought new life to research in a challenging area. However, this method only has the potential to solve the problem when the extremal configuration is unique, so in other cases we need new techniques.

In this talk we will discuss some methods for Tur\'an problems due to various authors, that incorporate some algebraic ideas. In particular we will present two new results: one gives bounds for the Tur\'an numbers of projective geometries, and another gives a general bound for the Tur\'an number of a hypergraph in terms of the number of edges that it contains.