Maximal k-Wise ℓ -Divisible Set Families Are Atomic

Let F⊂ 2^{[ n ]} such that the intersection of any two members of F has size divisible by ℓ. By the famous Eventown theorem, if ℓ= 2 then | F| ≤ 2^{⌊ n / 2 ⌋}, and this bound can be achieved by ‘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, for infinitely many ℓ, Frankl and Odlyzko constructed families F as above of much bigger size 2^{Ω ( n log ℓ / ℓ )}. On the other hand, in 1983 they conjectured that for every ℓ there exists some k such that if any k distinct members of F have an intersection of size divisible by ℓ, then | F| ≤ 2^{( 1 + o ( 1 ) ) n / ℓ}. We completely resolve this old conjecture in a strong form, showing that | F| ≤ 2^{⌊ n / ℓ ⌋}+ O(1 ) holds if k is chosen appropriately.