On systems of finite sets with constraints on their unions and intersections

AbstractLet F be a family of subsets of an n-element set. F is said to be of type (n, r, s) if A ∈ F, B ∈ F implies that |A ∪ B| ⩽ n − r, and |A ∩ B| ⩾ s. Let f(n, r, s) = max {|F| : F is of type (n, r, s)}. We prove that f(n, r, s) ⩽ f(n − 1, r − 1, s) + f(n − 1, r + 1, s) if r > 0, n > s. And this result is used to give simple and unified proofs of Katona's and Frankl's results on f(n, r, s) when s = 0 and s = 1