Intersecting Families are Essentially Contained in Juntas

A family J of subsets of {1, . . . , n} is called a j-junta if there exists J ⊆ {1, . . . , n}, with |J | = j, such that the membership of a set S in J depends only on S ∩ J . In this paper we provide a simple description of intersecting families of sets. Let n 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 1-junta (which by the Erdős-Ko-Rado theorem is a maximal intersecting family): for any such intersecting family A there exists an element i ∈ {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.