A proof of Alon-Babai-Suzuki's conjecture and multilinear polynomials

Let K = { k 1 , k 2 , ? , k r } and L = { l 1 , l 2 , ? , l s } be disjoint subsets of { 0 , 1 , ? p - 1 } , where p is a prime and F = { F 1 , F 2 , ? , F m } be a family of subsets of n such that | F i | (mod p ) ? K for all F i ? F and | F i ? F j | (mod p ) ? L for i ? j . In 1991 Alon, Babai and Suzuki conjectured that if n ? s + max 1 ? i ? r k i , then | F | ? n s + n s - 1 + ? + n s - r + 1 . In this paper we prove this conjecture.