Constrained versions of Sauer's lemma

Let [n]={1,...,n}. For a function h:[n]->{0,1}, [email protected]?[n] and [email protected]?{0,1} define by the [email protected]"h(x,y) of h at x the largest nonnegative integer a such that h(z)=y on [email protected][email protected]?x+a. We consider finite VC-dimension classes of functions h constrained to have a width @w"h(x"i,y"i) which is larger than N for all points in a sample @z={(x"i,y"i)}"1^@? or a width no larger than N over the whole domain [n]. Extending Sauer's lemma, a tight upper bound with closed-form estimates is obtained on the cardinality of several such classes.

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