The Junta Method for Hypergraphs and the Erd\H{o}s-Chv\'{a}tal 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 Erd\H{o}s-S\'{o}s `forbidding one intersection' problem and the Frankl-F\"{u}redi `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 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. We apply our method also to the 1974 Erd\H{o}s-Chv\'{a}tal simplex conjecture, which asserts that for any $d < k \leq \frac{d}{d+1}n$, 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 ${{n-1}\choose{k-1}}$. We prove the conjecture for all $d$ and $k$, provided $n>n_0(d)$.