Proof of an Intersection Theorem via Fourier Analysis

Abstract

Let p ≤ ½ and let μp be the product measure on {0,1}n, where μp(x) = p∑xi (1 − p)n − ∑xi. Let A ⊂ {0,1}n be an intersecting family, i.e., for every x, y ∈ A there exists 1 ≤ i ≤ n such that xi = yi = 1. Then μp(A) ≤ p. The proof uses discrete harmonic analysis.