Shattering-Extremal Set Systems of VC Dimension at most 2

We say that a set system F ⊆ 2[n] shatters a given set S ⊆ [n] if 2S = {F ∩ S : F ∈ F}. The Sauer inequality states that in general, a set system F shatters at least |F| sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly |F| sets. In this paper we characterize shattering-extremal set systems of Vapnik-Chervonenkis dimension 2 in terms of their inclusion graphs, and as a corollary we answer an open question about leaving out elements from shattering-extremal set systems in the case of families of Vapnik-Chervonenkis dimension 2.