Intersecting Families are Essentially Contained in Juntas

A family J of subsets of {1,...,n} is called a j-junta if there exists J subset of {1,...,n}, with vertical bar J vertical bar = j, that the membership of a set S in J depends only on S boolean AND J. In this paper we provide a simple description of intersecting families of sets. Let 11 and k be positive integers, with k < n/2, and let A be a family of pairwise intersecting subsets of {1,...,n}, all of size k. We show that such a family is essentially contained in a j-junta J, where j does not depend on n but only on the ratio k/n and on the interpretation of 'essentially'. When k = o(n) we prove that every intersecting family of k-sets is almost contained in a dictatorship, a l-junta (which by the Erdos-Ko-Rado theorem is a maximal intersecting family): for any such intersecting family A there exists an element i is an element of {1,...,n} such that the number of sets in A that do not contain i is of order (n-2 k-2) (which is approximately k/n-k times the size of a maximal intersecting family). Our methods combine traditional combinatorics with results stemming from the theory of Boolean functions and discrete Fourier analysis.