Alon–Babai–Suzuki’s inequalities, Frankl–Wilson type theorem and multilinear polynomials

Let K = { k 1 , k 2 , … , k r } and L = { l 1 , l 2 , … , l s } be subsets of { 0 , 1 , … , p − 1 } such that K ∩ L = 0̸ , where p is a prime. Let F = { F 1 , F 2 , … , F m } be a family of subsets of [ n ] = { 1 , 2 , … , n } with | F i | ( mod p ) ∈ K for all F i ∈ F and | F i ∩ F j | ( mod p ) ∈ L for any i ≠ j . Every subset F i of [ n ] can be represented by a binary code a = ( a 1 , a 2 , … , a n ) such that a j = 1 if j ∈ F i and a j = 0 if j ∉ F i . Alon–Babai–Suzuki proved in non-modular version that if k i ≥ s − r + 1 for all i , then | F | ≤ ∑ i = s − r + 1 s ( n i ) . We generalize it in modular version. Alon–Babai–Suzuki also proved that the above bound still holds under r ( s − r + 1 ) ≤ p − 1 and n ≥ s + max i k i in modular version. Alon–Babai–Suzuki made a conjecture that if they drop one condition r ( s − r + 1 ) ≤ p − 1 among r ( s − r + 1 ) ≤ p − 1 and n ≥ s + max i k i , then the above bound holds. But we prove the same bound under dropping the opposite condition n ≥ s + max i k i . So we prove the same bound under only condition r ( s − r + 1 ) ≤ p − 1 . This is a generalization of Frankl–Wilson theorem (Frankl and Wilson, 1981 [2]).