Small Doubling, Atomic Structure and ℓ-Divisible Set Families

Let (Formula Presented) be a set family such that the intersection of any two members of F has size divisible by ℓ. The famous Eventown theorem states that if ℓ = 2 then (Formula Presented) (Formula Presented), and this bound can be achieved by, e.g., an ‘atomic’ construction, i.e. splitting the ground set into disjoint pairs and taking their arbitrary unions. Similarly, splitting the ground set into disjoint sets of size ℓ gives a family with pairwise intersections divisible by ℓ and size 2^[n/ℓ]. Yet, as was shown by Frankl and Odlyzko, these families are far from maximal. For infinitely many ℓ, they constructed families F as above of size 2^{Ω(nlogℓ/ℓ)}. On the other hand, if the intersection of any number of sets in (Formula Presented) has size divisible by ℓ, then it is easy to show that (Formula Presented). In 1983 Frankl and Odlyzko conjectured that (Formula Presented) holds already if one only requires that for some k = k(ℓ) any k distinct members of F have an intersection of size divisible by ℓ. We completely resolve this old conjecture in a strong form, showing that (Formula Presented) if k is chosen appropriately, and the O(1) error term is not needed if (and only if) ℓ|n, and n is sufficiently large. Moreover the only extremal configurations have ‘atomic’ structure as above. Our main tool, which might be of independent interest, is a structure theorem for set systems with small ‘doubling’.