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… CONTINUE READING