Forbidden sparse intersections

Abstract

Let n be a positive integer, let 0 < p p ′ 1 2 , and let ℓ pn be a nonnegative integer. We prove that if F , G ⊆ {0, 1} n are two families whose cross intersections forbid ℓ-that is, they satisfy |A ∩ B| = ℓ for every A ∈ F and every B ∈ G-then, setting t := min{ℓ, pn − ℓ}, we have the subgaussian boundwhere µp and µ p ′ denote the p-biased and p ′ -biased measures on {0, 1} n respectively.