Shattering-extremal set systems of small VC-dimension

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S=F\cap S:F\in \mathcal{F}$. The Sauer inequality states that in general, a set system $\mathcal{F}$ shatters at least $\mathrm{\mid }\mathcal{F}\mathrm{\mid }$ sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly $\mathrm{\mid }\mathcal{F}\mathrm{\mid }$ sets. We characterize shattering extremal set systems of Vapnik-Chervonenkis dimension 1 in terms of their inclusion graphs. Also from the perspective of extremality, we relate set systems of bounded Vapnik-Chervonenkis dimension to their projections.