The junta method for hypergraphs and the Erdos-Chvatal simplex 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 problems such as the Erdos-Sos 'forbidding one intersection' problem and the Frankl-Furedi '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 Erdos-Chvittal simplex conjecture, which asserts that for any d < k <= d/d+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-1)(k-1)) We prove the conjecture for all d and k, provided n > n(0)(d). (C) 2021 Elsevier Inc. All rights reserved.