Some special vapnik-chervonenkis classes

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Abstract

For a class C of subsets of a set X, let V(C) be the smallest n such that no n-element set FX has all its subsets of the form AF, AC. The condition V(C) <+∞ has probabilistic implications. If any two-element subset A of X satisfies both AC = Ø and AD for some C, DC, then V(C)=2 if and only if C is linearly ordered by inclusion. If C is of the form C={∩ni=1 Ci:CiCi, i=1,2,…,n}, where each Ci 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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Partially supported by National Science Foundation Grant MCS-79-04474.