Abstract

Numerous problems in extremal hypergraph theory ask to determine the maximal size of a k-uniform hypergraph on n vertices that does not contain an ‘enlarged’ copy H⁺ of a fixed hypergraph H. These include well-known problems such as the Erdős ‘forbidding one intersection’ problem and the Frankl-Füredi ‘special simplex’ problem.

In this talk we present a general approach to such problems, using a ‘junta approximation method’ that originates from analysis of Boolean functions. We prove that any (H⁺)-free hypergraph is essentially contained in a ‘junta’—a hypergraph determined by a small number of vertices—that is also (H⁺)-free, which effectively reduces the extremal problem to an easier problem on juntas. Using this approach, we obtain, for all k in the range C to n/C, a complete solution of the extremal problem for a large class of H’s, which includes the aforementioned problems, and solves them for a large new set of parameters.

Based on joint works with David Ellis and Nathan Keller