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 widthω_h (x, y) of h at x the largest nonnegative 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 ω_h (x_i, y_i) which is larger than N for all points in a sample ζ = {(x_i, y_i)}₁^ℓ 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.