Shattering-extremal set systems from Sperner families

Abstract

We say that a set system F⊆2[n] shatters a given set S⊆[n] if 2S={F∩S:F∈F}. The Sauer–Shelah lemma states that in general, a set system F shatters at least |F| sets. We concentrate on the case of equality and call a set system shattering-extremal if it shatters exactly |F| sets. Here we discuss an approach to study these systems using Sperner families and prove some preliminary results based on an earlier algebraic approach.