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Studia Scientiarum Mathematicarum Hungarica
Authors: Dániel Gerbner, Nathan Lemons, Cory Palmer, Balázs Patkós, and Vajk Szécsi

A pair of families (F, G) is said to be cross-Sperner if there exists no pair of sets F ∈ F, G ∈ G with FG or GF. 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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