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Numerous problems in extremal hypergraph theory ask to determine the maximalsize of a $k$-uniform hypergraph on $n$ vertices that does not contain an`enlarged' copy $H^+$ of a fixed hypergraph $H$. These include well-knownproblems such as the Erd\H{o}s-S\'{o}s `forbidding one intersection' problemand the Frankl-F\"{u}redi `special simplex' problem.  We present a general approach to such problems, using a `junta approximationmethod' that originates from analysis of Boolean functions. We prove that any$H^+$-free hypergraph is essentially contained in a `junta' -- a hypergraphdetermined by a small number of vertices -- that is also $H^+$-free, whicheffectively reduces the extremal problem to an easier problem on juntas. Usingthis approach, we obtain, for all $C<k<n/C$, a complete solution of theextremal problem for a large class of $H$'s, which includes the aforementionedproblems, 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 simplexconjecture, which asserts that for any $d < k \leq \frac{d}{d+1}n$, the maximalsize 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$n>n_0(d)$.

The Junta Method for Hypergraphs and the Erd\H{o}s-Chv\'{a}tal Simplex Conjecture