Tightness of the dual VC upper bound for extremal classes

Determine whether the upper bound vc^*(C) ≤ 2·vc(C)+1 for extremal concept classes C is tight for all values of the VC dimension; specifically, ascertain whether for every integer d ≥ 1 there exists an extremal class with vc(C)=d and vc^*(C)=2d+1.

Background

The paper proves a new relation between VC and dual VC dimensions for extremal classes: vc^*(C) ≤ 2·vc(C)+1, significantly improving the classical Assouad bound for general classes.

While examples show tightness for small d (e.g., d=1) and for certain geometric constructions (e.g., hyperplane arrangements achieving vc^*(C)=d+1), the authors explicitly leave open whether the 2·vc(C)+1 bound is sharp for all d.