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If

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is a fixed hypergraph, then for two
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-free hypergraphs
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= ( V, E 1 ) and
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= ( V, E 2 ) we define their
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-free distance by the number of subhypergraphs of
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= ( V, E 1E 2 ) which are isomorphic to
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, and we denote this number by
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(
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,
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). For a collection
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of hypergraphs the
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-free distance of two
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-free hypergraphs (that is
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-free for all
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ε
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) is
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. In this paper we will obtain exact results on the maximum distance of K r -free graphs and Sperner systems and prove upper and lower bounds on the maximum distance of trees.

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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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