VC-saturated set systems

The well-known Sauer lemma states that a family F subset of & nbsp; 2([n]) of VC-dimension at most d has size at most & nbsp; sigma(d)(i=0)((n)(i)). 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 4(d+1), and thus is independent of n. (c) 2022 The Author(s). Published by Elsevier Ltd.& nbsp;