Abstract
For a class C of subsets of a set X, let V( C ) be the smallest n such that no n-element set F∋ X has all its subsets of the form A ∩ F, A ∈ C . The condition V( C ) <+∞ has probabilistic implications. If any two-element subset A of X satisfies both A ∩ C = Ø and A ∋ D for some C, D∈ C , then V( C)=2 if and only if C is linearly ordered by inclusion. If C is of the form C={∩ n i=1 C i:C i∈ C i , i=1,2,…, n}, where each C i is linearly ordered by inclusion, then V( C)⩽n+1 . If H is an ( n-1)-dimensional affine hyperplane in an n-dimensional vector space of real functions on X, and C is the collection of all sets { x: f( x)>0} for f in H, then V( C)=n .
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