A pair of families (F, G) is said to be cross-Sperner if there exists no pair of sets F ∈ F, G ∈ G with F ⊆ G or G ⊆ F. There are two ways to measure the size of the pair (F, G): with the sum |F| + |G| or with the product |F| · |G|. We show that if F, G ⊆ 2[n], then |F| |G| ≦ 22n−4 and |F| + |G| is maximal if F or G consists of exactly one set of size ⌈n/2⌉ provided the size of the ground set n is large enough and both F and G are nonempty.
Ahlswede, R. and Daykin, D., An inequality for the weights of two families of sets, their unions and intersections, Probability Theory and Related Fields, 43 (1978), no. 3, 183–185. MR 0491189 (58#10454)
Daykin D., 'An inequality for the weights of two families of sets, their unions and intersections' (1978) 43Probability Theory and Related Fields: 183-185.
Daykin D.An inequality for the weights of two families of sets, their unions and intersectionsProbability Theory and Related Fields197843183185)| false