On Subsets with Cardinalities of Intersections Divisible by a Fixed Integer

Abstract 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.