On Subsets with Cardinalities of Intersections Divisible by a Fixed Integer

If m(n, l) denotes the maximum number of subsets of an n-element set such that the intersection of any two of them has cardinality divisible by l, then a trivial construction shows that m(n,l)≥2 [n/l] For l= 2, this was known to be essentially best possible. For l ⩾ 3, we show by construction that m(n, l)2^−[n/l] grows exponentially in n, and we provide upper bounds.