| Popis: |
Let f be a set mapping from a set X into the power set of X. A subset A of X is said to be free for f if x ∉ f ( y ) ∀ x , y ∈ A and x ≠ y . For cardinals η and θ, we say that f satisfies the intersection condition C ( η , θ ) if | ⋂ f ( x ) : x ∈ Y | θ for every subset Y of X of cardinality η. We prove that if f : ω 1 ⟶ P ( ω 1 ) satisfies the intersection conditions C ( ω 1 , ω ) and C ( ω , ω 1 ) , then f has an infinite free set provided that the cardinality of every splitting family in [ ω ] ω is greater than ω 1 . Under MA + ¬ CH , such a function has an uncountable free set. In the above statement if one of the conditions C ( ω 1 , ω ) or C ( ω , ω 1 ) is dropped, then f need not have a free pair. However, if C ( ω 1 , ω ) and C ( ω , ω 1 ) are replaced by C ( ω , ω ) , then f has an infinite free set without having the cardinality condition for the splitting family. |
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