Affiliation:
1. 1 Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences P.O.B. 127 Budapest H-1364 Hungary
2. 2 Central European University Department of Mathematics and its Applications Nádor u. 9 Budapest H-1051 Hungary
Abstract
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.
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