Constrained versions of Sauer's lemma

Let [n] = {1, . . . , n}. For a function h : [n] → {0, 1}, x ∈ [n] and y ∈ {0, 1} define by the width ωh(x, y) of h at x the largest non-negative integer a such that h(z) = y on x− a ≤ z ≤ x+ a. We consider finite VC-dimension classes of functions h constrained to have a width ωh(xi, yi) which is larger than N for all points in a sample ζ = {(xi, yi)}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.