The VC-Dimension of Sperner Systems

Abstract

A family F of subsets of a finite set X shatters a set D X, if the intersections of the members of F with D coincide with the power set of D. The maximum size of a set shattered by F is the VC-dimension (or density) of the system. P. Frankl , is a Sperner system if for every two members of the system none contains the other (i.e., F is an antichain in the boolean lattice). A chain is a sequence of sets A i [m], 0 i n such thatA chain is complete if n=m and |A i | =i for every 0 i m.If D is a subset of [m] the trace of F on D is