VC-saturated Set Systems

The well-known Sauer lemma states that a family F⊆2[n] of VC-dimension at most d has size at most ∑i=0dni. We obtain both random and explicit constructions to prove that the corresponding saturation number, i.e., the size of the smallest maximal family with VC-dimension d≥2, is at most 4d+1, and thus is independent of n.