Proof of an Intersection Theorem via Graph Homomorphisms

Let 0 ≤ p ≤ 1/2 and let {0, 1} be endowed with the product measure μp defined by μp(x) = p |x|(1 − p)n−|x|, where |x| = ∑ xi. Let I ⊆ {0, 1} n be an intersecting family, i.e. for every x, y ∈ I there exists a coordinate 1 ≤ i ≤ n such that xi = yi = 1. Then μp(I) ≤ p. Our proof uses measure preserving homomorphisms between graphs. One of the fundamental questions first studied in extremal graph theory is the question of bounding the size of an intersecting family of sets. The most basic theorem in this vein is the Erdös-Ko-Rado theorem ([2] ) that states that if k ≤ n/2 and F is an intersecting family of k-subsets of {1, . . . , n} then |F | ≤ ( n−1 k−1 ) . The theorem we present here is the analogue of the EKR theorem in the setting of the discrete cube endowed with the product measure. This useful theorem, and several generalizations thereof has been proven and reproven in several papers (see e.g. [4], [5], [1], [3]), but curiously enough none of these proofs seem to be related to the one we present here which relies in a mysterious way on a decomposition of the n dimensional torus into 1-dimensional circles. Here is the main theorem: Theorem 1.1 Let 0 ≤ p ≤ 1/2 and let {0, 1} be endowed with the product measure μp defined by μp(x) = p (1 − p), where |x| = ∑ xi. Let I ⊆ {0, 1} n be an intersecting family, i.e. for every x, y ∈ I there exists a coordinate 1 ≤ i ≤ n such that xi = yi = 1. Then μp(I) ≤ p. Before proving the theorem we must introduce some notation. All graphs G considered in this note will come endowed with a probability measure μG defined on their vertex set.