On Mod-p Alon-Babai-Suzuki Inequality

AbstractAlon, 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 + krwhere kris the maximal element of K. Let $$\mathcal{F}$$ be a family of subsets of an n-element set. Suppose that (i) |F| ∈ K (mod p) for each F ∈ $$\mathcal{F}$$ ;(ii) |E ⋂ F| ∈ L (mod p) for each pair of distinct sets E, F ∈ $$\mathcal{F}$$ . Then $$\left| \mathcal{F} \right| \leqslant (_{{\kern 1pt} s}^{{\kern 1pt} n} ) + (_{{\kern 1pt} s - 1}^{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n} ) + \cdot \cdot \cdot + (_{{\kern 1pt} s - r + 1}^{{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} n} ).$$ 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.