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. [ABSTRACT FROM AUTHOR]