Families of finite sets in which no set is covered by the union of two others

AbstractLet fk(n) denote the maximum of k-subsets of an n-set satisfying the condition in the title. It is proven that f2t − 1(n) ⩽ f2t(n + 1) ⩽ (tn)(t2t−1) with equalities holding iff there exists a Steiner-system S(t, 2t − 1, n). The bounds are approximately best possile for k ⩽ 6 and of correct order of magnitude for k >/ 7, as well, even if the corresponding Steiner-systems do not exist.Exponential lower and upper bounds are obtained for the case if we do not put size restrictions on the members of the family (i.e., the nonuniform case)