The Junta Method in Extremal Hypergraph Theory and Chvátal's Conjecture

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 open problems such as the Erdős matching conjecture, the Erdős-Sós ‘forbidding one intersection’ problem, the Frankl-Füredi ‘special simplex’ problem, etc. 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 complete solution of the extremal problem for a large class of H's, which includes all the aforementioned problems. We apply our method to the 1974 Chvátal's conjecture, which asserts that for any d<k≤[Formula presented]n, 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 ([Formula presented]). We prove the conjecture for all d and k, provided n>n₀(d).