A proof of Alon–Babai–Suzuki's conjecture and multilinear polynomials. (January 2015) - Hwang, Kyung-Won
Abstract: 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.