On Mod-p Alon-Babai-Suzuki Inequality

Alon, Babai and Suzuki proved the following theorem: Let p be a prime and let K , L be two disjoint subsets of {0, 1, . . . , p − 1}. Let |K | = r, |L| = s, and assume r(s − r + 1) ≤ p − 1 and n ≥ s + kr where kr is the maximal element of K . Let F be a family of subsets of an n-element set. Suppose that (i) |F | ∈ K (mod p) for each F ∈ F; (ii) |E ∩ F | ∈ L (mod p) for each pair of distinct sets E, F ∈ F . Then |F | ≤ ( ns )+ ( n s−1 )+ · · · + ( n s−r+1 ). They conjectured that the condition that r(s − r + 1) ≤ p − 1 in the theorem can be dropped and the same conclusion should hold. In this paper we prove that the same conclusion holds if the two conditions in the theorem, i.e. r(s − r + 1) ≤ p − 1 and n ≥ s + kr are replaced by a single more relaxed condition 2s − r ≤ n.