The junta method for hypergraphs and the Erdős-Chvátal simplex conjecture

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-Sós ‘forbidding one intersection’ problem and the Frankl-Füredi ‘special simplex’ problem.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 C<k<n/C, a 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.We apply our method also to the 1974 Erdős-Chvátal simplex conjecture, which asserts that for any d<k≤dd+1n, the maximal size of a k-uniform family that does not contain a d-simplex (i.e., d+1 sets with empty intersection such that any d of them intersect) is (n−1k−1). We prove the conjecture for all d and k, provided n>n0(d).